I had a problem with the Mac Mini heating up when using flash for streamed videos. And I wondered about mounting it vertically, to better radiate heat.
First, though, I obtained a fan app that allows you to set temperature limits and fan speeds for all of your chosen temperature points.
That allowed me to monitor temperatures from multiple heat sensors inside the Mac Mini, valuable information. It does get hot in there.
So, even with that software turned off, mounting it vertically (using two bookends, or as one suggestion says, a napkin holder from any local store with a kitchen section) actually lowered the average temperature for the same usage by 10° centigrade, or 50° Fahrenheit.
SO many fan control apps are available, some for free, some for $15.00 or so. The one I settled on was Temperature Control Pro, by download for $15.00. A free version without some features is also available.
I really like it, and it works well to prevent Mac Mini and MacBook Pro fan over-running and overheating.
I used two bookends for a while, but went out and bought a clear plastic napkin holder from a local store for $3.99 and it works perfectly, and looks cool.
The commercial vertical holders I found were around $50.00 each. Such a ripoff.