Gary, Nope, when an incandescent bulb burns out the fillament that causes it to glow breaks and thus no current flows.
I can't absolutely say for sure about fluorescent bulbs, it may be that a very small amount of current flows through the electronic ballasts when the bulb dies, but I'd guess not. Of course if there is no bulb present no current flows, doesn't matter what kind of bulb you were thinking of putting in there. Tom Fowle