Does putting an unlocked phone on a no-contract plan lock the phone in any way? I bought a factory unlocked Samsung Galaxy S5 from eBay recently, and signed up with Ting.
(Was on Virgin Mobile with a different phone, switched carriers and phones at the same time)

When you're bringing your own device, Ting does ask for the IMEI to do a compatibility check, to make sure the phone will work on their networks (T-Mobile and Sprint), but it's optional, and after that, they only ask for your info and the Ting SIM card info (if you got one, which I did).

So I'm wondering: is my phone now locked to Ting or something similar?
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.