I have an application running on a server that's being backed up by tape. The application handles large amounts of business critical information on a daily basis. In this case how often should I be verifying that the tape backup and the SQL backup on the tape succeeded?
As often as practical!
Most backup software will produce a log of some kind, and has an option to verify the backup contents against the source when the job completes. At a minimum you should check the logs daily to ensure that the backup and verification phases completed without errors.
You should also test your disaster recovery plan on at least an annual basis, preferably every 6 months if loss of this data would cause loss of business.
Since you are backing up to tape, you should also ensure you are following the drive manufacturers guidelines for cleaning the tape drive. Also, ensure you follow the tape manufacturers guidelines for replacing tapes - typically a tape is good for about a year before it must be replaced.
You should include the "verification" step in your backup software. You should also do actual restore tests (quarterly, annually, whatever makes sense for you) - always remember that "performing backups" is not your job - "restoring data when disaster strikes" is your job, and "taking backups" is only a prerequisite step.