When it comes to plasma burn in, it very much depended on the manufacturer AND the product level. Low end 720p samsung plasmas had burn in issues, but those were still a bit earlier. The same generation Panasonic plasmas didn't have burn in issues. Granted, this is from displaying the same content over and over in a retail environment, but still.
I've got a 12 year old panasonic and an 9 year old and neither have had burn in issues. I even used the older one as a PC monitor for a few years.
As for OLED burn in, it seems it is on a per panel basis. Some people get it very quickly and it ruins the TV, others don't get it at all and don't see what all the fuss is about. My brother in law is the LG rep at his local Best Buy and I've asked him about burn in and he says it can happen, but you have to "abuse" your TV. Watching something with a ticker for hours and hours on end, static images that don't change for hours. Even if you do that and watch a movie or use the scrubber like on most plasmas, you will remove whatever retention you get.
Just like with Plasma people confuse burn in with image retention. One is permanent, the other isn't.
Don't do these things and you probably won't have issues.