what are the implications of installing APC on a VPS running multiple Drupal sites?
One thing to keep in mind is that APC will require quite a bit of memory if you have multiple sites because each file will be kept in memory, even if it is in fact exactly the same file, just with a different path.

So you need to watch and tune the amount of memory that apc uses/can use. By default, it's 32MB, that's often too less for even a single Drupal site if you have many contrib modules. The problem is that if APC has not enough memory, it will always have to delete files from the cache. If it has much to less (e.g. not even enough for all files in a single page request), it will make your site slower than if you hadn't used APC at all.

VPS often don't have a lot of memory, make sure that you have enough for your sites. You could consider merging them into multi-sites to save memory.