@lewm It all depends on the initial quality of your tap water. Distillation is effective at removing suspended solids (which you certainly don't want to be deposited in your grooves), and reduces dissolved salts (otherwise desalination of sea water would not work). When tap water dries, it does not leave metal ions on the record: they will be left as salts.
I've taken to using an air still to make my own distilled water, and when I look at what is left in it at the end of a run I am very glad it isn't being left on my LPs. My tap water comes from a deep drilled well and goes through a filter cartridge that gets changed every three months, but even so... It's instructive to let a puddle of tap water dry out on a sheet of glass and look at the residue afterwards. That's what will be left in the grooves.