If Palpatine were to survive, he would have had to survive three things:
The fall down the shaft
Here's a picture of that fall:
I can't even begin to speculate how long that shaft goes, but I'd hazard a guess it's at least a mile1. Palpatine is quite frail, so it's unlikely he could have survived. But let's say he did.
That chute leads straight to the Death Star's main reactor. According to Wookieepedia, the Death Star is powered by a Hypermatter reactor, which is powered by particle annihilation and releases enormous amounts of energy. Even if the Death Star reactor was 99% efficient (Which it almost certainly isn't), that's a ton of energy being converted into heat and/or radiation. It's a lot for an old man to bear for a non-trivial amount of time - he's down there for as long as it takes Luke to drag his extremely heavy father from the Emperor's throne room down to the shuttle bay, before:
The Earth*-Shattering Kaboom
That's the explosion Palpatine would finally have to survive. Note that it leave absolutely no debris, which means the explosion atomized the Death Star, and everything inside of it. Even if Palpatine could have survived the fall and the reactor, he's dust by now.
He did survive, at least in C-Canon. In 1991, Dark Horse comics published the first installment of the Dark Empire series, where we learn that Palpatine kept a stockpile of clone bodies lying around in case he was killed. It takes him six years to pull off, but he apparently managed to send his consciousness across space and into one of these bodies, whereupon he re-takes control of the remains of the Empire, and eventually takes Luke to be his new Sith Apprentice (Because some people never learn). So the answer to your question is yes, Palpatine survived (Or his consciousness did, anyway - that's a much bigger philosophical question I don't want to get into). At least he did until December 21, 20122.
With the recent shake-up in the Star Wars EU, Dark Empire is no longer canon. SO whether or not Palpatine survived is now in the hands of J.J. Abrams.
My bet is on "no".
1 As Dacio points out in comments, we know the shaft is no more than about 80 km (About 50 miles) deep
2 Holy crap guys, the Mayans were right