I am the executor of my mothers estate in Fairfax County Virginia. May I distribute money to heirs before the commissioner of accounts signs off from probate?

No, you have to wait until everything is competed and signed off... because until it is... something can come up to delay or prevent it happening.

SOME money, as long as there is plenty left, is usually allowed to be distributed before probate is over. Ask the court. Part of probate is waiting for any creditors to come forward, and if there was no will, any possible heirs.