And you know what ASS U ME does - - -
What happens depends on the construction of the lamp.
The brass socket style have craft paper type insulator between the brass shell and the internal socket. Overheat and it burns. Usually no fire - but now there is no electrical insulation - and the brass, having been overheated, looses it's temper - gets soft - and now the pressed together shell gets loose, and it is easily knocked apert - so the shell now shorts to the exposed terminal on the socket - and the lamp becomes "live"
If it is an open lamp, that's generally the extent of the damage - untill someone touches it and gets a (possibly serious) shock.
If the lamp is an enclosed fixture, snug to the ceiling or wall, the insulation in the fixture breakes down - the insulation in the socket
- if brass- deteriorates as above, with the same result - except it can short to ground - possibly blowing the fuse - possibly establishing an arc which can ignite anything flamable - including, possibly, the wire insulation in the box.
The high heat can also melt or warp plastic shades and difusers - and depending on the design, the plastic can contact the hot bulb - causing either fire or toxic smoke.
In a best case scenario, the overheated socket simply looses contact pressure on the center contact, causing the lamp to flicker or not work. The high resistance connection - if left powered on for too long, can severely overheat and, again, POSSIBLY cause a fire or damage to house wiring in the box.
At the very least - the socket/fixture itself is damaged and requires replacement.
Replacing with a ceramic (not plastic) socket can REDUCE some of the problems, again depending on the fixture design.
GENERALLY ceramic sockets are rated higher than brass (or steel) and ALWAYS higher than plastic (which should really be outlawed).