First, its' important to understand that the tax laws are federal, and are above and ignore state court orders no matter what they say.
According to tax law, the only parent who is entitled to claim a child as a dependent is the parent where the child lives more than half the nights of the year (183 or more nights). If that is you, then only you can claim the child no matter what the court order says. You file your tax return, say the child lives with you all year (or most of the year, whatever, as long as it is more than half), and say there is no court order to be followed. You will be allowed to claim your child. If your ex e-files first and claims the child incorrectly, that may block you from e-filing, so just print your return and mail it in. The IRS will investigate the duplicate claim and award the dependent to whoever can prove the child lived with them more than half the year.
Your ex can't claim the child unless you give your ex a signed release form. Even with the form, your can can only claim the child tax credit and not EITC or head of household filing status because those things stay with the parent who has majority actual custody, and can't be waived, transferred or shared.
If you have a court order that says you must sign such a release form to allow your ex to claim the child, that is between you, your ex and the court. The IRS won't care if you don't sign the release, they will award you the dependent. The court might care if you don't sign the form when you are supposed to, so be careful about ignoring a court's order unless you have a good reason and good legal advice.
