I watch video regularly using movist, VLC and QT player, either from files locally stored on the rMB or via wifi network from another computer nearby. Have never noticed the computer getting especially hot, and it handles playback flawlessly. Sometimes it stutters but that's only when I'm far away from the wifi router and the transfer rates are too low - nothing to do with the rMB itself but everything to do with the (currently) poor internet/wifi setup I have at home.
In terms of watching things online, I regularly go on youtube and have had no issues with anything playing smoothly, including a large 4K video that someone once posted a link to here. No noticeable overheating with youtube either. I removed flash so (I think) this forces youtube to use the HTML5 version of its video player instead, this used to be the case a long time ago at least. Perhaps this is more efficient than if youtube uses flash? I don't know what kind of player Amazon Instant Video uses, but if there are different format options offered, you could try using different ones to see if it improves things.
