Well, in my case, the command I use reads something like:
wget --auth-no-challenge -q -O- --http-user=my_username --http-passwd=my_password 'https://ipv4.tunnelbroker.net/nic/update?hostname=123456'
It turns out this is safe to run at any time. It just tells you that there is no change ("nochg"). On the other hand, you should not run it too frequently
Edit: If you just go the advanced tab on the tunnel details page, it will provide a URL to use.
Edit2: I think the other part of the equation (at least for me) is to omit the specifying the local IP address in the configuration. Linux will use your local address as needed.