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.

I have a MacBook that I bought in 2009 (2.4GHz Core 2 Duo, 4GB 1067 MHz DDR3), I started with Leopard and had the following experience with upgrades over the past 5 years:

• Snow Leopard - noticed a performance improvement
• Lion - noticed significant slow down of all OS X features/applications, was seriously considering buying a new MacBook
• Mountain Lion - massive speed improvements immediately after upgrade. Start-up time for all applications has improved by a factor of 10, especially for Safari and iTunes. I used to click an application icon and wait for 10 to 30 seconds, now it opens immediately. Obviously, Apple OS X engineers have worked hard to tune and optimise 10.8 for all hardware, even the old the stuff.

Anecdotally, Mountain Lion seems to actually be faster than Lion. There aren't that many new features in Mountain Lion, which means Apple's engineers had more time to analyze and tune its performance. We know, for example, that they specifically optimized Safari scrolling to make it smoother than it used to be. The Versions feature is also faster (or rather, it does more of its work invisibly in the background).

Apple made some improvements to the graphics subsystem, including the OpenGL stuff that can boost performance by up to 10 percent, depending on what you're doing.

Since Snow Leopard, Apple has been relying much more on using the graphics cards in non-traditional roles, including OpenCL, so this sort of performance boost won't just be limited to programs that render 3D. Any app that uses Core Image will get advantages from increased graphics performance, for example Pixelmator, which makes use of the GPU for image editing.