I skimmed alot of this, so sorry for any repeat info:
Plasmas are better than LCDs and LEDs, IMO. The deeper black, faster processing, and high amount of color make them ideal for a movie buff like, myself.
Burn-in is very rare, these days, and limited to very exteme cases. IR, or image retention, as Village Idiot mentioned, is more common, but I've still not had too many issues with my Samsung HP-T4264. The one problem I've had is I fell asleep with a DVD on and the image of the menu was retained. However, I did the whitewash method and it was gone within an hour.
Never had any other issues with IR, however. I will say that Samsung advertises that their TVs are already broken-in, but I suggest using the break-in settings on any new plasma, anyway. The pixels burn hotter and brighter the first 100 or so hours of use, and it's easier to get IR or burn-in during those hours.
I also suggest that you go to the AV Science boards and get the optimal settings for your device (assuming you don't have it professionally tuned). The picture quality is a world of difference.
Some people on here will argue with me, others on AVS forums agree with me (so you know it's likely somewhat even), but I prefer Sammy over Panny. I did months and months and months of research before making my investment, and I felt the pictures looked much better on the Sammy and the features were much better (don't know about current sets. This was '08), and thus I bought the samsung.
I haven't had a single issue, and I've been perfectly happy with my set.
On the other end of the spectrum, if you get a cheap set, do expect IR.
FWIW, I bought a 32" Vizio plasma last year, I believe, and it has the worst IR I've ever seen. No burn-in, luckily, but it has IR even when opening credits pop on the screen for a few seconds. The now discontinued vizio plasmas used LG displays, so I would recommend against those, personally.