cleaning question

I just got a really nice stainless steel strained. Would it hurt to throw it in the dishwasher after use? Or would that put too much chlorine on it and affect my next beer.

IMHO - It should not cause any issue to run it through the dishwasher.  Any chlorine will evaporate off while it dries between uses.  And you’re going to rinse again before using it anyway.

Paul

+1  Dishwashers do a great job on SST and glass items.