The parent where the children live, who has actual physical custody more than half the nights of the year, is the only parent who is automatically entitled to claim the children as dependents. Regardless of any divorce order. Federal tax law trumps any state law or court order.
The only exception is if you have a divorce that was signed in 2008 or earlier, which gives him the right to claim the children as dependents in certain years, and which contains certain specific language. If that is the case, then he can claim them as dependents by attaching a copy of the divorce decree to his tax return.
Otherwise, only you are entitled by the IRS to claim your children as dependents. He cannot claim anything unless you give him a signed form 8332 which releases the dependent exemption for certain years. Form 8332 can be signed for one year at a time or it can be signed for several years in advance. If you gave him a signed form 8332 several years ago and that form gives him the right to claim the children this year, then it is too late to retract the form for this year. However, if he does not have a signed form, then he can't claim anything.
His only recourse would be to go to the Family Court judge and ask the judge to order you to sign the form or hold you in contempt. I don't think he realizes this is the case, because if he is behind on his child support, going to the Judge to ask for an order against you would be putting himself at greater risk. So I think he is either ignorant, or bluffing.
However, the risk is on you. You may want to consult with your family law attorney. If you claim the children as dependents and he also does, then the IRS will start an investigation of both of you. You would need to provide proof that the children lived with you.
Also important: even if he is allowed to claim the children because of a pre-2009 divorce order or a signed form 8332, he can only claim the dependent exemption and child tax credit. The ability to qualify for head of household, claim EIC, and the dependent care credit, always remain with the parent who had custody more than half the nights of the year. Those tax benefits can never be waived, transferred, or released to a noncustodial parent. In TurboTax, you would enter the children, state that they lived with you more than half the year, and that you are transferring the exemption with the signed form 8332. TurboTax will give them the proper status to claim head of household and EIC without claiming their dependent exemption.
