I think when I rang to cancel my phone line with BT, they gave me a date a few days later. My ISP gave me a cancellation date about 2 weeks later. I asked BT about this, and they said it would carry on working until the ISP cancelled it.
Also, at home, we have a separate line for broadband (for complicated reasons). Due to a complete screw up transferring from BT to Caudwell Communications, the line rental stopped being paid on this line. It had been cut off for around 3 months. We never noticed, as we don't use it for voice calls!
I think generally when you cancel the line, it will carry on working. But you don't know for how long - could be a couple of days, could be a year. If BT need your DSLAM card for a new connection, they'll just unplug you.
It's not advisable, for those reasons. Plus, if you ever had a fault BT would do sweet FA about it.