I own 2 Roku Premieres and both are connected to LG 4K televisions and are as out in the open as you can get them. They are placed on top of the entertainment center, but whenever I would stream a movie through a premium service, I would have an overheating issue.
I tried every suggestion that Roku (and other posts) have suggested. Turn off the device, reboot the device, make sure that you have good HDMI cables, make sure the device is out in the open, etc.
Allow me to also preface what follows with this: I also own 2 Roku Express devices connected to other 1080p TVs in the house that have never experienced an overheating issue. Also, they are in use more than the Premiere devices.
Last weekend I was streaming "The Suicide Squad" on HBOMax and 1 hour in to the movie. It began overheating. Keeping in mind that my Express devices have never overheated, just for the heck of it, I went into Settings > Display Type and changed it from "Auto Detect" to "1080p TV."
The Roku immediately stopped complaining of overheating and I was able to stream the rest of the movie without issue. I have not received an overheating warning since.
Maybe the chipset that Roku is using for the premiere is not able to handle 4k streaming yet, maybe there needs to be a firmware update. I do not know. All I know is that as soon as it was set to 1080p it was fine.
Do with this information what you may, but I would like to hear if anyone else has tried this or if they have their own fix for this problem.