A buddy of mine was an armorer for a large police department.
He used to joke that his 870 spare parts would "Fit into a shoe box, with enough room left over for a burger and fries".
With that said, it couldn't hurt to keep spare:
Extractor, extractor spring, and plunger.
Firing pin and spring.
Off hand I can't think of anything else that might break, that's user replaceable.
The reason about 95% of all American law enforcement use the 870 is, they just don't break very often, even when abused as only street cops can.
I'd spend my spare money on shells, not parts I'd likely never use.
The top sources of Remington parts are:
Brownell's
http://www.brownells.com
Gun Parts Corporation
http://www.e-gunparts.com