Technically, Tower Bridge sits just east of the Congestion Charge boundary. The official zone ends at the A1211, which runs roughly along the line of the Inner Ring Road, and the bridge itself is outside that ring.
But here’s the twist, and it’s a big one: the approach roads—like Tower Bridge Approach and the roundabout at the northern end—are still in the zone. You can literally be parked on the bridge and be fine, but the moment you cross onto the pavement (or the approach slip road), you’re liable.
So if you drive across the bridge from south to north, you’ll enter the zone around the Minories junction. Yes, you read that right: the bridge itself is safe, but the road leading to it is not. Classic London logic.