Can I do a SQL Server backup to a tape drive on another server?

A. No, not with built-in SQL tools. SQL will only dump to local tape devices. If you could find an NT driver that would make a remote tape drive look local then this would work as SQL just uses standard i/o calls. I don't know of such a driver at the moment.

You can always dump the SQL database to disk locally (or to a network share - see networkdrivebackup.txt for details) and then back that up to tape.

Finally, you CAN do what you want with a 3rd party backup tool that has a SQL agent. Examples are BEI Ultrabac, CA Arcserve-IT, Veritas BackupExec, Veritas NetBackup, Legato Networker and IBM ADSM - I'm sure there are others.
These put SQL dumps (via a standard named-pipe or VDI interface) onto an NT tape, potentially as part of a complete server backup and not just SQL. Obviously dumping across a network is usually slower than doing it locally.