Can riding without a chain damage a bike?
What harm could it possibly do? The cranks are held firmly in place by the bottom bracket; the back wheel is held firmly in place in the drop-outs. The chain is completely non-structural: all it does is transmit power from the pedals to the rear wheel.

The only issues I can think of are


make sure you know how to remove and replace it, obviously;
I pretty much guarantee that you'll fall off at least once because of this. You know, as an intrinsic part of riding a bike, that putting forward pressure on the pedals has various effects and you use those effects subconsciously to control the bike. Now you're riding a bike where putting forward pressure on the pedals just spins them with no resistance. But, hey, you're practising tricks so I assume that falling off isn't a deal-breaker. :-)


Benedikt Mokroß adds


You're going to hit your knee on your stem a lot. Been there, done that.