You didn't mention a maintenance level but make sure you're at the latest maintenance level (e.g. 5.1.1). Thats probably not the problem though. The problem sounds like TSM is unable to access the media for whatever command you're running (in this case, move media, restores, and reclamation).
At the TSM admin prompt if you do a "help anr1144w" you can get more information about the error. In the text of that message it says "explanation: during move data processing for the indicated volume, a required volume cannot be mounted". Again indicating that the TSM server software is unable to mount the media for some reason.
So a few questions.. when issuing the commands does your tape library attempt to mount any cartridges?
when issuing "q vol" you say everything looks good and is in read/write state, but how about "q vol f=d"? Look at the field "Volume status". Are the volumes "on-line"? or some other status? If their status is unavailable or offsite, that could explain why it's unable to mount the volume.
Do a "q act" for the timeframe that you're attempting any tape related commands. Are there any other errors logged during that period?
ultimately you need to determine why the required volume cannot be mounted. I can think of a number of reasons why this may occur but often many of them have additional pieces of information in the activity log indicating why. I would probably even run a "dsmadmc -consolemode" and watch that screen while i perform the above command to see what gets logged there.