Some tape drives will tell you when they need a cleaning, many won't. If not, you should clean it after a certain number of uses, depending on the type of drive. (Or perhaps "once a month" if that's close enough...)
Do you get those error messages every time you use the tape drive, or just with one tape? If it's only with one tape, obviously throw that tape out.
Also - many people back up to tape but never bother to test their tapes. It's worth checking occasionally to see if your tape backups really do work. (Carefully! You don't want to accidently restore old files from a tape over newer files on your disk drives.)
These problems seems to be with your Tape Drive (Exabyte), even it also display there are some fatal errors in writing in very speecific Error Block 6406, so make sure your disk is corrupted or not, if it needs to be cleaned. Try sending this ASC: 0x3 (<vendor unique code 0x3>), ASCQ: 0x2, FRU: 0x0 to vendor.