I would like to use Visual Studio Team Services for Work Items tracking, and use our on-premise installation for source code, builds, etc. Is this possible? If so, do the TFS versions need to match?
I'll assume you're using TFVC, if you're using Git the answer will be a bit different.
I guess it could work, but you would lose the ability to link work items and changes together.
You could just reference the work items in commit messages, for example:
Fix #123: No longer crashes.
But the association would not work.
If you had on premise build and source, everything would work, it would really be like using TFS with a separate Work Item software, I know some people use Jira instead.
As there will be no integration between the 2 the versions needn't match.
If you moved your source control to VSTS, you can keep your build agents locally, there's a cost to it IIRC, but it is possible.