My sons father does not want to be on his birth certificate but we have already done a DNA test and waiting for the results, if it turns out he is his son does the that mean he has to go on it or can he just refuse?
Also my son is only 1month old and his dad has only seen him a few times since his been born and I want to relocate,but only to another city not state, am I able to do this without his permission if he is not on the birth certificate or if he chooses to go on? Either way?

... if it turns out he is his son does the that mean he has to go on it or can he just refuse?
...
The rules for whether the father are put on the birth certificate will vary from state to state. Those rules will specify when or if the father of an out-of-wedlock child are put on a birth certificate.


... I want to relocate,but only to another city not state, am I able to do this without his permission if he is not on the birth certificate or if he chooses to go on? either way?

If and when he gets a custody/ visitation order you have to abide by the terms of that order. Until then, you can take the child to another state if you want.

ANY question on law needs to include your general locale as laws vary by area.

If you do move and he files for custody, you may be forced to return. Whether he's on the birth certificate or not matters little. If you go after him for child support (and you may be forced to if you use public assistance), then he will be declared the legal father.