Will running out of fuel oil damage 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:
What is the difference between home heating oil / fuel oil and diesel?