Can you post the running config of both? Could it be that the ACL for the tunnel on the remote site doesn't allow the subnet on which the DNS sits? You said you can ping from a PC at the remote site. Can you ping the DNS server specifically? Without further information, I'm wondering if your remote PC is only trying to resolve through the Internet DNS, not the internal zone.
Here's something to try, though. Open ASDM then go to Monitoring | Logging | View. . . (Set at Logging Level: at debugging first, though.)
While you've got that open, try to resolve something through DNS from the PC. If it's an ACL, you might see a deny. But you'll at least get a hint of what's happening. You also might see whether it's going out to the Internet DNS instead of the tunnel.
If it does look like it's going across the tunnel, open logging on the other side of the tunnel, using ASDM again, to see what it's doing when the DNS segments come across.
Finally, if it shows that the UDP 53 segments make it through the firewall and across the tunnel successfully, you can always use Wireshark (or similar) on the DNS server to see what's coming in and whether it's even replying.
ArchiTech89
CCNA Routing & Switching, CCNA Security
MCITP, MCTS
Berlin, Germany