My personal favorite live performance from TV would probably be Nirvana's performance of Lithium on the 1992 MTV VMA's. It's my favorite because of some of the things that happen during it:

Krist Noveselick Cracks his head with his own bass when tossing it in the air and it lands on him.

and the best part is when Dave Grohl jumps onto the mic and yells "Hey Axl, Hey Axl, Where's Axl, Hey Axl" over and over.

here's the video if you want to see it:

what other favorites are there