Absolutely nothing. The government can't coerce religions to perform ceremonies. The right to marriage is not a religious right, it is a civil right, and will be provided by civil authorities.

For example, in Massachusetts anyone can become a marriage officiant by applying to the governor. There's a form; permission is always granted. As a result, I have performed two marriages.

The better question is, what happens to a civil official who declares that it is against their religion to perform their civil duty?