Will I damage anything by not replacing my clicking bottom bracket?
Depends on what's clicking.  If the bottom bracket itself is making the noise then no hurry to deal with it.  The bearing may be slowly failing but nothing else will be damaged and the bearing will probably give you plenty of warning before it fails completely -- if it ever does.

But if the noise (and generally an accompanying sensation) is due to the crank arm slipping on the crank spindle then continuing to ride on the bike without tightening the crank bolt will quickly destroy both spindle and arm.

(I get the impression that often cartridge BBs click because the cartridge is simply loose in the frame.  Tightening the cups may be all that's needed to silence it.)

(Another thing to be aware of is that pedals often make a clicking noise, and it's difficult to distinguish this from BB noise.)