We wish we knew the answer to this
Some older BT-Vision boxes are designed to only work if you’re paying BT for broadband service and line rental.
Some BT-Vision boxes will still play and record Freeview broadcast TV without a BT broadband but won’t stream Internet TV.
Some BT-Vision boxes will work quite happily without a BT broadband. For example, Currys are currently, September 2014, advertising BT-Vision boxes with no subscription and no tie-in to any particular ISP (See here for details).
To further complicate matters, some people have BT-Vision on contract with BT and when your BT Broadband contract expires you will be expected to return the box to them.
So it’s a bit complicated and we can only suggest that you contact BT and ask about your particular box and contract.
Sorry we couldn’t give you a simple answer. It’s almost as if BT were intentionally making things complicated to stop people moving to a different ISP