I do the same. chlorine will evaporate from the source water overnight if left in an open container. simple, inexpensive de-chlorination...I fill my spray bottle with tap water that I have let sit out for a few days. Not sure if it has any benefit but I do it anyway. I have forgot to fill my sitting container and used fresh tap water also.
that's not from chlorine, its from dissolved solids in the water, simple dechlorination will not remedy that.The only bad thing i know of related to dissolved salts in water you mist your tarantula with is that they precipitate onto the glass, thus making it hard to clearly see your Ts after several months.