Is it correct to use this phrase to mean "be useful or beneficial"?
Yes, but using it like that on the phrase "to helping people communicate" is very strange. Usually, a person is the object of "doing good" like "Cutting back on sweets didn't do me any good" . Or with no subject "Better recordkeeping has done a world of good".
Or you could say "It does no good to only floss once a month". "Milk does a body good" was an advertising slogan some years back,
Your sentence would be better if it were "It does no good at helping people communicate." Or "It is no good at helping people communicate."
A "do-gooder" is someone who tries to help people, but it's a condescending term. It suggests that the person speaking is scornful of the effort (A villian might call Superman a "do-gooder"), or that the person is doing good more for attention than the actual good act, or that they are trying to be helpful, but failing.
Yes, "do harm" is also a real phrase; the Hippocratic oath in English is "First, do no harm". but you don't use it as much, probably because "harm" itself is already a verb, while "good" is not. But you could say "That heatwave did serious harm to our crops".