It is interesting that your dome light still works, to the extent that it might be interesting that your cigarette lighter still works - they are electrical, and in the same car. However, they are not part of the same circuit with which you are having the problem.
I side with Rocky in that it
has happened in the history of mankind that a bulb's filament has broken in such a manner as to short to ground with very little resistance/load. This happens in large bulbs with beefy filaments, and in a frequency to be well under statistical significance. A burned out bulb, especially in something as small as a dash lamp, is going to be an OPEN.
What is causing your fuse to blow is almost certainly a special type of "short" circuit - one that is a DEAD SHORT to ground, with no load or "work" being done by the electrons on their way to ground. The circuit is very simple, with very little going on, and a component notorious for failure along the way.
Admittedly, some of these points might be subtle, and not immediately apparent to someone who has not spent decades working on cars. Other concepts, however, should be as basic as they come - what happens when a bulb burns out, what makes the fuse in one circuit blow repeatedly, and the nature of a dead short...
...especially when the one posing the questions is a
trained engineer who sees fit to belittle the unwashed masses, both in
threads on the General Board, and his sig line [edit: Yes, I see you just now changed it]. Imagine us being foolish enough to think a car should be equipped with a dipstick for the transmission.
I was planning to bite my tongue, and just offer polite help, until I read your followup questions.
Perhaps your dash lights are out because of the presence of a "caveman stick" in your car somewhere...