The Apple TV uses the latest iOS versions (unless you've managed to keep an older model from updating).
As such, the Apple TV will require a semi recent version of iTunes to talk with on your computer.
So if you allow your Apple TV to update, then you'll want to keep iTunes up to date as well.
The current versions of iTunes require Mac OS 10.8.5 or later. And so that is the minimum at this date.
As time moves forward, iTunes will require Mac OS 10.9.x and then 10.10.x, and then 10.11.x, etc.
So if you go old, then don't let anything update, or you'll eventually find yourself with something that no longer works together.
If it were me, I'd be shopping for something that can run El Capitan as a minimum. Preferably something supported by the upcoming high sierra.
On the flip side, if you don't mind running your iTunes library on Windows, then a core2duo will probably be fine. Most core2duos can run either Windows 10 or Windows 8. And iTunes will be supporting both versions of Windows much longer than they'll be supporting even El Capitan. For example, last I checked, Apple still supported iTunes running on Windows Vista, but required Mac OS 10.8.5 or later.
Update:
Apparently, Apple has changed the requirements since the last time I looked.
It appears that for full functionality, you'll need at least Windows 7 or newer, or Mac OS 10.10.x or newer.
Personally, here's what I have done... taken a used PC ($40 on Craigslist) and installed iTunes on it as a media server. With Windows 10, you'll have support for a long time and not need to worry about updates killing your media center arrangement.
Or, if you like tinkering, throw Mac OS on that cheap PC. My Hackintosh has proven to be a very stable Mac. I'm happy with it. And I've moved my iTunes library back over to Mac OS with it. It's stable, reliable, and I have no complaints.
I have it set up for dual booting. So I have the best of both worlds on it. I'm actually starting to like it more than my Mac Pro (which sucks cause I have thousands of dollars invested in my real Mac and only $40 and time invested in my Hackintosh).