Can a heatsink protect a PYNQ-Z1 board from damage? If so, which one should I use?
Adding a heat-sink won't hurt your board or any other board connect to it. 
Also fitting one which is too large is not an issue either, as long as it does not touch anything but the top of the chip.

Before you go though all the trouble have a look at the applications which cause it to heat up so much and what speed they are running. It could well be that the circuit you are planning to program into it will not heat it up.

There is a power estimation tool in Vivado which can give you some idea of the energy it will use before you program the board.