Inside your gas tank is something called a fuel pump, which is responsible for sending the fuel from the tank to the engine, where it is burned for propulsion. This pump relies on the liquid gasoline itself to keep it both cool and lubricated. When your fuel is too low, it can overheat. If this happens too often, you risk premature failure of the fuel pump, and that is a costly repair to undertake. Failing to replace a fuel pump can lead to poor performance and, eventually, your car just won't start at all (via AutoGuide.com). 

If you live in a cold climate, there is another reason letting your fuel run too low is dangerous; condensation can cause fuel lines to freeze. Frozen lines can prevent gas from reaching your engine, leading to a car that won't start. Not a fun thing to wake up to on a frigid morning just because you didn't want to stop for gas on the way home the night before. While we totally get how very much stopping for gas isn't something you want to do after a busy day, these are a few very good reasons to put on your big kid pants and do it anyway.