Yes, permission from the biological father is necessary. And, leaving without court permission puts a strain on the court order for child support, too. Finally, you'll want to involve Alaska legal courts in the situation to protect your child so follow all the rules to do this right.
In Arkansas there is a presumption that if the custodial parent wishes to relocate that they will be allowed to do so, and the Arkansas child support order would still be valid. If the father does not comply with the order, you can either file a motion directly in the Arkansas court to enforce the order, or you can ask the Alaskan Title IV agency to pursue it on Alaska's behalf (though it will often be quicker if you file the action yourself.
With that said, you need to have someone review your order to ensure it does not contain language that restricts your ability to take the child out of Arkansas, and the father does have the option of going back to court to try an rebut the presumption that you be allowed to relocate.
You need permission from the other parent or the court when leaving the country with your child, but unless your flight is going to another country first then you won't need a passport or the father's permission.
As to why the Court would care, the Court cares because it is there to determine and protect legal rights of parties, as well as to uphold its dignity by ensuring that its orders are followed.
Sign up to receive a 3-part series of useful information and advice about child custody law.