it gets confusing because HDTV means several different things if you think of it how I outlined above it's a lot easier to understand.
Burn in only effects certain Display technologies like CRT and Plasma. It doesn't effect stuff like LCD or DLP.
Even still CRT tech has come a long way and unless you totally abuse your display you wont have to worry about burn in. That means have your brightness/contrast properly tuned (not cranked way up like it comes out of the box) and avoid playing for long periods of time with static elements on the screen. Most modern games are much better than older games. Games with solid powerbars and bright text that never moves are much more likely to cause burn in than transparent huds that come and go.
As for 1080i support most displays will "support" 720p and 1080i and simply convert one or the other to it's native resolution. The Best displays will support both without conversion (much like your CRT PC monitor supports multiple resolutions). But as Redwolf said the 360 can output in either format so whatever your display supports will be just fine
I hope that cleard some things up