What are the laws if my son's birth certificate has a father listed that I am sure is not the real father? The person's name that is on the certificate is aware of this and wants it to remain this way and the real father has no interest in our son and has not in almost 2 years (my son is 3).

The law varies by state. If the person who is listed as father was your husband at the time of marriage, then most make him the legal father automatically unless challenged.

Doyou live with the person who is lsted as the father, if not is he paying child support ?

No he is not my husband and yes he does live with us. I guess my concern is that if the real father all of the sudden decides he wants to be in his life will it create a big problem legally?

Depends on how soon the real father acts.

Has the listed father signed an acknowledgement of paternity?

No he has not signed an acknowledgement.