My longest is when watching the World Cup, about 2 hours. Not sure how hot it should be, but I did feel the phone being warm.
I have mine set to stay on while charging, and some days, I'll leave it plugged in all day so that I don't get screen-blanking issues during any critical calls.
On most of those days of 6~8 on time - the screen can stay mildly warm.
When my screen has gotten hot, it's because I've been doing stuff that pushes the processor to top speed for those hours.
The LCD panel itself should have a mean time between failure rating on the order of 10s of thousands of hours - the backlight should be good for some thousands of hours or even in 10s as well. These sort of displays are typically rated for much longer, but I've de-rated them here for conservatism and mobile use.[*]
However, when those components begin thermal conduction and in effect become heat sinks for the processor or the battery - THAT IS A PROBLEM.
I'd gladly elaborate - but only as requested. Basically - you can kill a panel with excess heat.
And - because you're interesting in other technologies such as SAMOLED, I thought I'd share this, somehow I wasn't paying attention to it.
Color e-ink is here. No backlight, very high power efficiency, the brighter the sunlight, the better it looks - the whole bit. Here's some links if you're interested.
It's very cool - just uses a form of light interference to achieve color, and 300+ dpi resolutions are possible. Right now, it seems that color saturation is a little low - but - give it time.
Qualcomm's Mirasol displays - no backlight required! Live Photo Gallery
How mirasol Displays Work: Micro-electro-mechanical Systems (MEMS) Drive IMOD Reflective Technology | mirasol