This may not be a dns issue
If you like you can set your dns manually in windows.
Here is a nice guide on it
keep your ip address
automatic (select "obtain ip address automatically) If this is not already selected this would also cause your problem.
Some common dns servers are
126.96.36.199 (Level 3)
You can find a full list here
I do however suspect it is a redirect issue. You may have security software blocking this. Next time you are at a hotel please do the following
1. Run a command prompt
2. type in ipconfig
3. find your interface default gateway
4. type the default gateway into the browser (usually 10.0.0.X or 172.16.0.X or 192.168.X.X)
It should take you to the login page for your hotel.
Please post the results here and we can try for a permanent fix or if it is successful and you are happy with this solution. Than I wish you well.