Your name being on the mortgage has nothing to do with this in any way, form or fashion really. What matters is the name(s) on the property title/deed. Lets assume that your name is also on the deed.
Since you personally have no rental income or rental expenses from this property, you have nothing to report on any SCH E at any level. Your father will report 100% of everything on both his federal return and state return. However, when the property is sold that's a different story. It will be assumed that each of you received 50% of the proceeds from the sale.
I would highly suggest you both seek professional advice from a tax pro in the state of NJ. I would expect them to offer you two options:
1) If the mortgage holder will allow it, you do a quit claim deed on the property giving your dad 100% legal ownership. If your dad can not qualify for the mortgage on his own, the mortgage holder most likely will not allow it. But you don't know if you don't ask.
2) You and your father establish a partnership which requires all rental income/expenses to be report on IRS Form 1065-Partnership Return. With that setup, you can divide ownership in the partnership where your dad has 99% and you have 1%. (You can't have a 0% ownership partner. Must be at least 1%). Then depending on the laws of NJ, it might be possible for you to be a "limited" partner where you only receive a payout if certain conditions are met - such as requiring a certain income threshold to be exceeded before limited partners are paid.
So seek professional help on this so you two get things right the first time.
