"I see there are some that can display 4:3 material in its original aspect ratio, like the Zenith models."
I believe they all can
"Well, what does that look like? Are the left and right sides gray or black?"
Depends on the model's particular options, on my version, you have a choice of white, gray, and black. The exception is if its an HD feed then you can't control either the aspect ratio or the side bar color. On some manufacturers' models you can control these parameters on HD feeds too.
"Is this a surefire recipe for burn-in?"
Yeah if thats all you watch for months and months. I normally watch in Just mode which just expands the ends to fit in the 16:9 format and IMHO is very acceptable.