Will Mountain Lion make an older computer run faster or slower?
Faster. 

I have a Mid 2007 iMac 2.4 GHz Intel Core 2 Duo with 4GB RAM. This is the oldest iMac model that can officially run Mountain Lion (according to Apple's tech specs). 

I upgraded the first day and haven't noticed any slowdowns so far. Performance actually seems faster in Finder, Mail, Preview, etc. 

Safari stands out the most in terms of noticeable speed improvements.