My home heating oil is very low. I have 2 questions: 1. If I run out of heating oil can my tank/ furnace become damaged? 2. Can I use diesel fuel until I am ready to fill my tank?
If you run out of fuel, your burner will shut down, as there is a safety system that disables it if there is no flame when the burner is running. You are likely to clog your filter, so plan on replacing it if you do run out (it may be good to proactively replace it if you run very low, even if you don't run completely out). After replacing the filter, you'll need to press the reset button, and there may be a procedure to bleed air out of the line (I've seen the service tech do this on ours after the system shut down due to extreme cold clogging the filter with gelled fuel).
In other words, you'll need a service call after you run out, unless you are comfortable maintaining your system yourself.
My understanding is that diesel fuel has additives that are not needed for an oil burner, but will do no harm to the system. You are paying road taxes that will increase the cost, but in an emergency, that's probably not an issue (I keep a couple six gallon plastic Jerry cans exactly for this contingency because we have a driveway that is sometimes not passable with an oil truck in the winter).
Edit: a web search on "difference between fuel oil and diesel" seems to suggest that they are mostly interchangeable, with some minor exceptions -- a cetane additive may (or may not) be added to road diesel in some circumstances, but should not affect your oil burner negatively. Kerosene may be added to fuel oil (not sure why), but often isn't, so diesel would again be an OK substitute.
Here's one of the sites I read:
Standard diesel fuel and home heating oil are basically the same thing. Sometimes kerosene is added to home heating fuel to keep it from gelling. I actually put the cetane additive into my heating fuel for this reason due to my tank being above ground. It won't hurt anything to put 10 gallons of diesel into your tank to keep you going until you can get fuel delivered.
When I had an oil-fired heating system, I ran the tank dry on many occasions with no problem.
The outlet of my tanks was not at the bottom of the tank but maybe an inch higher. This kept sediment and water out of the filter.
The only problem I had with one system (and only one) was that the line needed to be bled. This was a time consuming (up to 30 minutes), messy and smelly process. This system had a long length of pipe horizontally from tank to boiler/furnace. Other systems (perhaps with shorter runs and more of a slope) had no need for bleeding, they just fired up immediately after the tank was refilled.
You might be able to use diesel - I'd check with the boiler/furnace manufacturer.