Does putting an unlocked phone on a no-contract plan lock the phone in any way?

No one locks your phone when you haven't purchased it from them with a contract - your S5 will stay unlocked when you use it with Ting or anyone else.

Of course, different carriers do need different APN settings etc. which will overwrite the existing ones in the phone - but this should normally happen automatically when you insert the new SIM. If not, then you might need to search for the carrier settings and do this manually (trivial to do anyway).

As for the IMEI check done by Ting, that is because in some cases phones may be banned from the network they're using (reported as lost, having an unpaid balance etc.) and in that case the IMEI gets blacklisted, which 3rd party resellers can't do anything about.