My boyfriend received nys court date papers here to my address but he does not legally live here if we send it back will it show that he does not

If the boyfriend does not live with you you can return the papers to the sender and inform them that they have not properly served your boyfriend. If he has notice otherwise (like service at an alternate address perhaps), then he has been served and needs to attend or have an attorney enter an appearance on his behalf. If he fails to answer or appear, he may ultimately need to provide proof that he was living at an alternate address and was improperly served. He should discuss this with an attorney because he might end up with a default against him that can lead to things like wage garnishments, suspended driver's license, etc.