It allows alot more dust to get inside the case, and also has the components sitting out without any protection... If there are children around, and they build up a static carge, they could discharge it into the PC if they touch something inside the case. I think static discharges are normally around 40 volts, MORE than enough to FRY electronic circuit board components. Along the same lines, if you eat or drink near your PC, a spill could result in a less than desired 'mess' in the case depending on how your 'computing area' is set up.
As far as running cooler than usual, if you have a case that overheats often, it would likely run cooler. What reason would you want to do this? What kind of system are you running? If you have an OEM system (HP, Compaq, Dell, etc. etc.) I wouldn't even bother poping the side off, it wouldn't likely help... Also, the right side of the case (where the bottom of the mobo is) won't do anything if you take it off, as the mobo tray pretty much covers the entire side (typically).