Are divorce decrees honored by the IRS? What if the child does not live with the parent, can that parent get full tax credit?

Only pre-2009 divorce decrees, but paying child support cannot be a condition in order to the child.  After 2009, the non-custodial parent can only claim if the custodial releases the exemption with a signed 8332 form which must be attached to the non-custodial parents tax return.

The IRS only cares about physical custody.  The parent where the child lived more than 1/2 (or greater part) of the tax year (more than 183 nights) is the custodial parent.

That does not mean that you can ignore any local court decree as the other parent can go to court to have it enforced and force you to issue the 8332 form or be in contempt, but that is a local family court issue, not a IRS issue.

See IRS Pub 17

https://www.irs.gov/publications/p17/ch03.html#en_US_2015_publink1000170876

Children of divorced or separated parents (or parents who live apart)

Post-1984 and pre-2009 divorce decree or separation agreement.   If the divorce decree or separation agreement went into effect after 1984 and before 2009, the noncustodial parent may be able to attach certain pages from the decree or agreement instead of Form 8332. The decree or agreement must state all three of the following.
    1.    The noncustodial parent can claim the child as a dependent without regard to any condition, such as payment of support.
    2.    The custodial parent won't claim the child as a dependent for the year.
    3.    The years for which the noncustodial parent, rather than the custodial parent, can claim the child as a dependent.

  The noncustodial parent must attach all of the following pages of the decree or agreement to his or her tax return.
    •    The cover page (write the other parent's social security number on this page).
    •    The pages that include all of the information identified in items (1) through (3) above.
    •    The signature page with the other parent's signature and the date of the agreement.

Post-2008 divorce decree or separation agreement.   The noncustodial parent cannot attach pages from the decree or agreement instead of Form 8332 if the decree or agreement went into effect after 2008. The custodial parent must sign either Form 8332 or a similar statement whose only purpose is to release the custodial parent's claim to an exemption for a child, and the noncustodial parent must attach a copy to his or her return. The form or statement must release the custodial parent's claim to the child without any conditions. For example, the release must not depend on the noncustodial parent paying support.