The only way really to test for high temps is to do something intensive on it, like a game.
As for screen burn, go to a black internet explorer page, that's all white. Then you can tell. Screen burn is probably a lesser issue than discoloration due to overheating. You can get a brown or green stain on the screen if it's overheating. Sometimes these will do that on a wireless charging pad. If left overnight, it can burn the screen.
For the mics and speakers, the easiest way is to call yourself from the phone, then use it on regular and on speaker phone. See how the sound quality is on both, on your phone and the new phone. Sometimes the regular mic doesn't work, but the speaker mic does, or vice versa. You have to have the other person talk into on both, and see if you can hear. Do it the other way around too, where they talk to you on your phone, and you make sure you can hear, on speaker and not.
Check both cameras as well - make sure the image isn't foggy or anything - nice and clear.
You don't see this go bad often, but you can also check the auto-screen dimming feature.
Speaking of wireless charging - if you or the seller has a wireless pad, you might want to bring and put the phone on it to make sure it charges. If you leave it on there for about 10 minutes, you should get an idea of whether it gets hot or not.