During heavier BitTorrent periods I can no longer resolve DNS addresses via my DrayTek Vigor 2820n router, however the DNS servers themselves can be pinged and sites can still be connected to by IP.

The only way to resolve the issue is to reboot the router.

What exactly is happening?

The router might be too taxed to handle doing DNS forwarding, you can fix that by getting it to give out the DNS addresses directly by setting the DNS servers you want to use on the LAN > General Setup page.

Different firmware versions might handle bittorrent / heavy traffic better than others as well, what firmware do you have on the router right now?

May be unrelated (I don't torrent) but I tend to set the "bandwidth limit" to fractionally less than the connection will handle at peak, seems to reduce problems with saturated link. I.E. if your maximum (tested) throughput is 6500/350 set the bandwidth to 6300/330.

This looks to be an issue with the DoS protection. I had them all enabled at their default values, but when switched off the problem goes away. Don't know which one is the issue.

I'm experiencing the same problem, I'll try switching the DoS Defense off but it'd be good to know exactly what is causing it as I'd like to switch at least some of the protection back on.

It would be interesting to know if you are using a fixed IP on the wan side.
If you get a dynamic address was that previously used by someone for P2P and all of their peers try to access the address you now have it would be hard to distinguish that from a DOS attack.
As you will be downloading only legal stuff ;-) I'd have thought a fixed IP address is worth trying if you really do want to use DOS, that way the peers have a fixed end point over time, less likelihood of sudden mass connection tests.