Will the coin spin forever or will it be de-accelerated by the gravitational force?
Assume there is no other force (such as friction), and say the coin is near the Earth and experiences the Earth's gravitational field, then with classical mechanics, the gravitational force only acts on the center of mass of the coin, and produce no torque, so the coin, if it is spinning initially, will remain spinning forever.
This is the rotational analogy of Newton's 1st law, which states that any object that's moving in constant velocity, without any external force, will remain moving in constant velocity.
If would like to elaborate on the answer give by Leo L.
Tidal forces have to be considered, too. Those will eventually lead to a rotation of the coin locked to its orbit, like moon. In this specific set up, however, I reckon that tidal forces are so small that they can be ignored for all practical purposes. The term „practical purpose“ here includes the moon which had enough time to synchronise its rotation.
The crucial aspect is the relation of the radius of an extended object to its orbit. For the coin it is much smaller than for moon and the timescale of synchronisation increases with a very high power of that ratio.
Elaborating on my own answer: tidal forces only apply when the body is not completely rigid but is deformable by the gravitational forces.
If the coin is considered rigid then you can do the maths and integrate all forces over all of the volume of the coin and the result is as described in the answer by Leo L.