Can users still log in computers at remote site when primary DC down?

If you still have the Site-to-site VPN link up and there is another DC in the same domain elsewhere and there is an alternative DNS server (maybe your other DC) that can till be reached, then yes this should still work, however users may also logged in with cached credentials depending on the network and if the clients can see other DCs to authenticate with and have access to other working DNS servers.