That question has come up a few times in different forms (e.g.,
Can water go stale? and
What E61 & Grinder would you buy for <$2K?), but I don't recall any blind taste comparisons of brew boiler versus fresh water. I'll pose my suggestion again for any owner of a moderate to large dedicated brew boiler espresso machine (e.g., Expobar Brewtus, Isomac Amica, Fiorenzato Colombina, or multi-group commercial machine):
Of course, there's an easy test: Assuming the machine is squeaky clean, draw eight ounces of water through the group and let it cool. Compare the sample blind against fresh water, also at room temperature. What is the difference in taste, if any?
I did have the Brewtus for a few months and never noticed an issue, but I didn't specifically compare the water drawn from a squeaky clean machine after it sat idle for n hours / 1 day / n days. The boiler material would probably make a difference, e.g., I would be surprised if a stainless steel tank imparted a taste after prolonged downtime, but a copper one might.
(Of course the simple answer is don't worry, drink more espresso.)