When two parents live together unmarried and share custody, only one parent can claim the dependent and all the tax benefits, and the other parent should not even list the child on their tax return. There is a way to "split" the dependent benefits when parents live apart and share custody (whether divorced, separated or never married) but these special rules don't apply to unmarried parents living together.
Unfortunately Turbotax is not clear on this. When your boyfriend entered your child's name and indicated that you had a custody agreement, and that you were claiming the child, the interview assumed that you were living apart. It gave your boyfriend certain of the split tax benefits of a dependent. He needed to say "no", there was no agreement to share custody (because that question only applies to legal court orders for divorced or separated parents) or even better, he should not have listed the child at all.
If you have decided that you will claim the child, you should print your return and mail it in. After your boyfriend's tax return is accepted and his refund paid, he needs to file an amended return to completely delete the dependent from his tax return. He will also have to repay any EIC or other tax benefits he claimed, including head of household status.
