No, it wont. I'm cleaning mine for decades with anything at hand (cotton shirts, napkins, etc) and not a scratch.
And of course there's the little fact that microfiber cloth is a recent synthetic thing. People used cotton and linen squares, or chamois leather ones if they felt fancy, to clean their glasses.
To clean glasses safely you basically need a soft, clean cloth. Cotton is totally fine. You could get away with a soft clean sponge, too. Or even a soft-ish piece of paper (which is what most disposable lens words are.)