My guess is that one of the door cutoff switches has popped loose and isn't opening the circuit when you shut the door as it should. Try manually depressing them see if you can get 'em to go off. But that's about as good as I can do without looking at it.
possible short to power or short to ground/
this is kinda of a lame answer but i had the same problem too my battery died about 4 times because of it. but my solution was i took all the buttons out of the doors and tailgate except driver and it has't done it since. it you take the switch out it acts as if the door is closed, you could use that as a test and see which one shuts off the light. hope i was some help
Looking for a Used Blazer in your area?
CarGurus has 1,050 nationwide Blazer listings starting at $950.
Search Chevrolet Blazer Questions
Chevrolet Blazer Experts
Related Models For Sale
Used Cars For Sale