That would be grammatically incorrect in English, and it would sound very strange.
While you could say "František does not do laundry," that wouldn't be an accurate translation of the Czech sentence, because the original includes the word "never."
You could also say, "František doesn't ever do laundry," or "František doesn't do laundry EVER" (for emphasis), but I don't think these two versions are accurate translations, either. We'd need a native Czech speaker for that.
I can't think of anything that would be better than "František never does laundry."