Imagine, hypothetically, that despite being told not to, someone keeps using a hair dryer and space heater in the same outlet at the same time, and thus tripping the circuit breaker. Will this have any detrimental long-term effects on the breaker, wiring, or anything else? Or is it fine to just keep resetting the breaker as many times as it takes them to learn?
The breaker is designed to heat up at about the same rate as the wiring in the walls. However it tends to cool off much quicker than wiring in walls. It's expected you are going to leave the breaker off and resolve the underlying problem, not reset it over and over.
So if it is frequently reset, the wiring gets hotter and hotter and the breaker is out of sync with just how hot it is. This can lead to a fire.
A typical hair dryer is 1500W or 1800W. Almost any space heater is 1500W. However many space heaters have a "low" setting in the 700W ballpark.
A 15A circuit has 1800 watts.
A 20A circuit has 2400 watts.
Neither one can support two 1500W devices.
Nonetheless, rather than blaming the user, you should consider that technology is meant to serve us, not the other way 'round. The user's requirements are reasonable, and the technology should meet it.
The real culprit here is a single 15A or 20A circuit supplying power in the bathroom, and possibly serving other rooms as well. It's true that NEC allows bathrooms to be served by one 20A circuit, but you are certainly allowed to have 2 circuits in a bathroom - and that's the best cure for 2 heat appliances at once.
However, even better would be to dump the dangerous space heater and get built-in 240V heating, such as a baseboard.
A simpler method would be to obtain a space heater with a low/high setting, and either educate the user on use of the "low" setting with the hair dryer, or hack the space heater so the "high" setting gives the "low" setting.
Breakers, like any other device, wear out. There are latches in them that will get rounded out and not lock in place when being reset. If the two devices you're referring to are plugged into the same outlet, the constant overloading will weaken the jaws of the outlet. Breakers don't trip immediately on an overload, they "cook" a little bit and tht overload can/will eventually damage the outlet.