After you leave the charcoal paste on your teeth for around 3 to 4 minutes to allow it enough time to bind to the teeth then you can rinse your mouth out several times.
After you rinse your mouth out of the charcoal paste several times then you should brush your teeth after you use the charcoal on your teeth.
So yes you do need to brush your teeth after using charcoal on your teeth but after rinsing your mouth out first.