What if a grandmother gives permission to someone not blood related to claim her grandchild on their tax return. Does she have the right to do that because her child is 17?

And what are the penalties considering the grandmother has a previous felony?

A grandparent has no rights to allow anyone to claim a child that doesn't qualify under IRS rules.

Can you explain a little more about the situation? Example: The grandmother has custody of the child and also lives with someone else who also provides for the child. In a case like that so long as the taz laws are followed then e nonrelative may use the deduction so long as they qualify.

Only the person who meets the requirements to claim the child can claim it. If the child is Qualifying Child of someone, no one else can claim the child.
Also nonrelative does not get EIC by claiming the child.

I would point out that a non-relative can claim a child as a dependent ONLY if the child lived with that person all year, and that person provided at least half the child's support. Also, the child must not be a "qualifying child" of anyone else - meaning the child was not in the custody of a relative, including parents or grandparents.

NO ONE has the authority to allow a non-relative to claim the child who does not meet the criteria specified under IRS rules.