Should I leave my computer on or turn it of when I take a short break?

If you are going for just a few minutes then just leave it on or send it to sleep. Shutting it down really isn't worth it in that case. Overheating shouldn't be an issue here. Sleep will save a lot of power. In fact, I can make my Mac's battery last a week or longer if I send it to sleep. As for the environment, it really isn't a significant effect for laptops but if you had a desktop that ran a large power supply then it would make a slight difference hypothetically. Overall, if you are just leaving for a little, sleep or leave it on. If you are going away for a day or two shut it down to save some power. And I believe it requires more power to stay on then startup.

if your worried about your computer over heating sitting idle for a few minutes then you have bigger issues.

for a few minutes no it probably will not save you any energy because it takes time and power to start up and shut down too, maybe put it in standby mode

If computer is not overheating (if it has good cooler and temperature is normal) it is better for it to stay turned on all the time. When computer is off and you turn it on, electricity comes into parts and when you turn it off it comes out (0/1)and doing that is "bad for computer health" as far as i know. Correct me if i am wrong.

Over-heating would imply heavy workload, not sitting idle. Though sitting idle is going to keep your CPU warmer than simply turning it off, naturally. If your computer runs hot on idle, you should check on services you have running eating too many resources and if you have proper cooling for your hardware.

If your BIOS allows it, you can also try hibernating. Hibernating saves your memory to disk and shuts your computer off, saving all your power and saving all your session.

Suspending (aka Sleeping) will keep your memory in the RAM so if the computer loses power it loses the memory, but it will save power in general. However, because of its volatility, Suspending/Sleeping isn't the best choice for longer sessions. Unlike Hibernation though, you can Suspend your computer and return it quicker than it takes to power the machine on from POST and MBR (usually).

As for power consumption, these are some rough percentages I pulled from thin air. :P Nothing statistical here, so don't quote me (but should give you the idea)

Off = 0% or 1% (if you have wake-on-lan enabled)

Hibernate = 1% - 10%

Suspend/Sleep = 5% - 20% (would depend on machine hardware)

On = 100% (obviously not 100% unless full load, but like I said, rough idea purposes)

Also, like with ANY device, mechanical or electronic, that initial startup requires a high level jolt of power to get everything powered up, so if you only need to leave it for a very short while, it might actually be better to just leave it running.

it's always been recommended to leave computer on not turn off and continually reboot , it is starting and booting that cause sudden bursts of heat and shut down more sudden cool down this effect of heating and cooling is not only hard on the hardware its hard on solder that holds wires pl;ugs and such together as well as circuit boards processors makes solder brittle crack break and even at times hot enough to melt ..

leave the computer run 24/7 after a bit it will hibernate and sleep it wont over heat just being on the auto maintenance and updates will happen on there own like they are supose to as well so your always up to date safe and secure .. here is how it works at worst leaving a computer laptop no matter on 24/7 rebooting one a month or so just to make sure ram is not fragment you might this is just a might at worst find a fan could need replacing from use turning the PC on and off many times and all to soon hard drives, video cards, processors , ram start showing early signs of wear grinding cable start to dry rot usb ports start to fail ..

leaving computer on wont make them over heat once not in use they sleep and cool down in a very short time,you can add thousands of hours of usefull life to computer hardware by keeping it clean dust free and just not killing it over and over with off and on use ..

this hold true even more with today new ultra books and tablets that are small slimline nothing plugs in everything is all solder to the mother board perminatly machines most glued tight shut not repairable or up gradable they are appliances you use till they stop then you dispose of them , most today at least win 7 have instant on technology built in no need to shut it down some of the machines dont even come witha off and on button ..