Nah, the armrests of the chairs fold in ;)
What’s funny is that with the tumblehome hull, the Zumwalt probably rolls more readily, which means that the bridge really will tilt a lot, especially during extreme maneuvers or if the ship takes a hit. So it’ll be more like the TOS Enterprise than one would think.
In the era of guided munitions that isn't as much of a problem but I still wonder why stability isn't as much of a concern has it has been until now.