I have a 6115 in a closet still with which I always had the monitor on top and never had a problem, there was sufficient distance between the top of the case and the motherboard (having opened it up to do an add-on card) that it is unlikely that was the cause.
As for the Mac Mini, I have to agree that you do want to have good airflow around the case as it doesn't use a fan. The motherboard on my 2011 Mac Mini died I believe because I had it in a cabinet resting flat. Fortunately, I had AppleCare and Apple replaced the motherboard for free. I have since bought a stand so that my Mac Mini sits vertical, and haven't had a problem since, knock on wood.
You can buy a stand for the Mac Mini on-line, mine is an aluminum arch with a slot in the middle where the Mini rests.