This question is far too broad to be practically answered in detail. If you were asking about specific dangers, that would be appropriate for this forum in my read of the question requirements, but you are asking something that invites opinion and cannot reasonably be responded to comprehensively because you haven't given us any details about the specific hazards you will encounter. Answering your question comprehensively would require a complete index of every possible hazard in every possible work environment. For this reason, my experience strongly indicates that you don't know enough about safely Operating the question you have plans to rent to even know if it's the best idea for achieving your project goal. Dangers aside, whatever it is you're digging, you may not even be aware that you might make more work and cost your self more by attempting to DIY than hiring a professional. Additionally, since you haven't asked about specific dangers, and since this strongly indicates you have done no research into training material for operating such machinery, I can't reasonably assume that you actually have the competency to build a proper jobsite barrier or to work around utilities or structures that might hurt someone beyond any barrier you do errect. Anyone encouraging you to proceed with your plan to operate this piece of machinery without adequate training to indicate a level of knowledge to safely operate such machinery would be, in many contexts, indirectly complicit in causing any incidents you may directly bring about. Knowing the dangers is not the same as knowing how to avoid them, and it is not enough to safely operate the machine.
A previous version of this answer suggested you should not operate such machinery as you indicated planning to rent, and cautioned that if you have to ask such a broad and basic question, you should strongly consider getting formal training first as this lack of basic understanding of the fundamentals of responsible Operating implies you are truly ignorant to these dangers, especially as you have not indicated this is a question for academic purposes. If you asked me personally, face to face, for my honest opinion, that is what I would tell you. Here, on this free-use forum with it's own culture and expectations, I won't speak from the heart, and instead here is some more generic advice... The greatest danger to the operator and any other person or thing is inexperienced Operators and the human error that results from inexperienced operating.
I won't outright tell you not to rent this and to hire a professional, but I will say that an open-ended question on a subject that has ample information in great depth directly answering this question available to you for free online really strongly indicates you have not researched the topic to the fullest extent a responsible operator might expect would be appropriate before even beginning to plan to rent and operate a piece of such equipment.
I will also tell you that if you have to ask such an open-ended question, and since this has been up for some time and you have not made it more specific despite the breadth of answers, you are strongly indicating that your lack of knowledge is the greatest risk to the driver, as well as everyone one and thing that could get near you - as such, I would strongly propose you suggest to yourself, 'if I have to ask such an open ended and naive question, not for academic purposes, I should strongly consider the fact that their is no reasonable and safe reason to proceed with my plans to operate this piece of machinery until I have training or a depth of knowledge that permits me to understand the basis of safe Operating."
When beginning to attempt to operate Heavy Machinery (even when they are referred to as "mini" or "light" models) it is best to keep the mantra "Crawl, Walk, Run" in mind, especially for inexperienced Operators who are concerned with dangers to their-selves as Operator and looking to prepare their-selves to rent the equipment. Asking "What are the dangers of operating a mini excavator?" is practically saying "I plan to rent an airplane, what are the dangers of flying one."
Since it is reasonable to assume you are an adult, and since in general on Stack Exchange if we don't see a nationality clarifying your citizenship, we assume you are a citizen of the USA, there is a strong cultural tendency to hold the sentiment "Nobody can tell you what to do! It's your god given right!" close to your heart. Other's will tell you, people do rent these all the time and it turns out to be a great time, but you are asking what the dangers are and the primary danger is proceeding with a plan to rent and operate one of these without training or understanding even the basic capabilities of the machinery, despite the fact that there are typically no laws against it in the USA.(I do suggest you check your jurisdiction for city ordinances and codes that may require licensed operators, but ususlaly this is not the case). For example:
Can a mis-move of the levers cause the bucket to break through the cage?
No, and that is a fundamental design regulation of the equipment you are planning to operate.
Nevertheless, if you do decide to proceed with your plan instead of hiring a pro, here is some reference material: here is the spec and Operator's Manual for a mid-size excavator (the big sister to the class you're proposing to rent). The operator's manual is old, but still the basic point of the rest of my post is the 1-2-1 "Operate Only If Qualified" clause. These manuals and the hazards they outline are 'must know model-specific material' per manufacturer's requirements for anyone operating heavy machinery (although federal regulations only allow enforcement of that requirement to employees; you're local or state jurisdiction may be subject to broader restrictions). Within this context, the manufacturer requires that you read the Operators manual (such as linked to above) prior to operating the machine, even if you are a "certified" operator - the regulations are, you still have to read it and know the information within prior to operating the machine. (Whether that is broadly and consistently enforced or even practical to broadly and consistently enforce is another topic). This is to say, failing to read and comprehend the entirety of the Operator's Manual that remains in the cab of whatever unit they drop off for you (regulations require that their must be an Operator's Manual in good readable condition), is the first and principle danger you will have a chance to avoid before you even put your key in the ignition.
Next, as you will see recommended by machinery manufacturers in the Operator's Manual, and as the Federal standard for Earthmoving machinery will require of those Operating such machinery during the course of their employed daily activities (but to which you as a private citizen not Operating this Machinery for any business purpose), you are told you must be trained by a competent trainer - remember, as a private citizen, their is no means to enforce this Manufacturer's requirement except where your local jurisdiction is subject to such regulations having ammended Federal requirements. If you choose to pursue training, 'adequate training' (as far as the requirements for meeting Federal regulations goes, which is a helpful benchmark for bare-minimum definition of competency for Operating Earthmoving equipment) would be achieved by either of the following: either by a competent, qualified, experienced trainer you're friends with, or by a professional trainer (by searching online for local equipment training facilities, or finding one through the recommendation of a local equipment rental company). It's worth the money if you have no experience with equipment like this, as it could save your life, or at the least it is insurance you won't destroy your driveway or similar incidences as you try to get the equipment to the area of work. Here is an example of the subject matter training material your course would cover, tailored specifically for the type of equipment you would be getting certified to operate.
Please keep in mind: As others will also tell you, your rental agency is insured against you or a damaged party suing them for their equipment's involvement in an unsafe incident. That shouldn't be encouraging. Their insurance doesn't cover you, and generally you should expect to be agreeing to indemnify the rental company for any damages to equipment or otherwise upon accepting receipt of the equipment. It's like renting a car. If you drive inebriated in a rental car, you still go to jail. With the rental car, you have to have liability insurance. For the equipment, if you don't have an individual umbrella policy, homeowners or rental policy that covers the types of incidents you might cause, it's all on you baby.
I think it's fair to assume you have no experience with heavy equipment. I would like to share some of my experience, but I want to talk to you about understanding the whole environment you'll be operating the equipment in, not just the hazards to you as the equipment operator because the chance of you getting hurt is far less than the chance of someone else getting hurt from something you've done with the equipment.
I am truly not trying to insult you. I am going to try and give you some perspective on just how deeply out of your depth you sound like you might be. I get it, you are an adult, and this could be a fun experience where no one got hurt, but you could also kill someone. Since I have no idea whether you are on a farm with no chance of someone random walking across your path, or if you are in a crowded neighborhood with a little old neighbor who might go searching frantically for her missing dog in the middle of the night and find herself at the bottom of a pit dug in your yard (real story, years of litigation), I'll assume the higher end of trouble just in case.
OSHA is bare minimum (see 1926.602 for reference). A number of people here have advised you to search through their tools and regulations. I have an OSHA 10 cert (twice), an OSHA 30, and over 80 hours of university level practical instruction on writing an employer safety manual for commercial construction, and two decades work experience. Many DIYers will disagree with me, but looking through OSHA should be the very lowest thing you look to if you want to learn how to safely and effectively operate equipment. OSHA does not prescribe how to operate safely, it is literally regulations describing activities you will be fined for. Looking to OSHA for learning how to safely operate heavy equipment is like memorizing speed limits of roads in your area. That's not going to make you a better, more defensive driver. I say this because it's extremely common to start reading through the OSHA website.
Similarly, ANSI Standards cover the benchmark for best practice. It is all still guidelines. It's not meant to tell you how to do it safely, it's meant to define 'safe operation' in detail in legally actionable terms. But, ANSI Standards would have you take an operator certification course - professionally taught, and licensed/accredited as competent and qualified instruction (usually that means that the manufacturer has trained and certified their trainers in the specific equipment they will train you on). I think you should get trained, but do you have to? I mean, if the company rents it to you, then no you don't have to, especially as a homeowner, but you should be aware that the issues aren't just the dangers of operating heavy machinery...
Please, before you put the key in the ignition, thoroughly and deeply understand the following:
You need to understand the hazards throughout your entire jobsite (your work area), including the dangers to the driver, if you truly want to work responsibly. For example, are you up to current best practice on protecting open excavations. At a much more basic level, do you know to locate utilities before you dig, and how far away you need to start digging with hand tools only? Are you aware of the personal/criminal negligence you are liable for in the event of harm even if you claim to not know you were doing something dangerous? Did you know in general throughout the USA, if someone sneaks into a jobsite even with all the barricades and safety precautions you think you have, if they hurt themselves you still can end up legally responsible? Tangentially related to operating the machine, and as much the responsibility of you as the operator as safe operation, are you sure you understand all the regulations that govern the work you are undertaking? Does your jurisdiction have temp erosion and sediment controls? You should look into the cost of how much you can get fined for violating runoff regulations. On a very basic level, do you know what kind of fuel your equipment takes? How about where to find the operators manual in the cab? In my opinion, you won't be competent to 'safely' operate an excavator until you know the answers to these questions.
What I'm saying is that there's a lot to worry about far beyond just "How tippy is this thing."
Now on the other end of things, heavy machinery are super fun to operate (until they become your job and then it's just a thing you do for work). They can also destroy anything you accidentally bump or run over in an instant, faster than you can react, including a life. Being hydraulic, they can also cause damage if parked improperly... and, fun fact, they will cause severe chemical burns if you get the hydraulic fluid on you!