What would cause an intermittent increase in water pressure? I Noticed that when our clothes washer is running that the water pressure increases enough that the toilet valve will not shut off and that water was dripping from the faucet with the valve closed. This faucet does not drip when the washer is not running.

You will need to do some investigating with this problem. I'd start with a temporary pressure gage installed on one of your cold supply stop cocks to a sink. Watch the pressure before , during and after a wash cycle. It should be about 40-45 psi all of the time with maybe a hiccup when the washer fills and shuts off. Can you tell if the faucet drips hot or cold, the toilet would only be hot.

Is there not a valve to keep back flow from happening normally in the washer ? Even if there is, a back flow connector could be placed at the outlet of water to the washer ( between faucet and water hose) this would stop any back flow of water from the washer into the house system.

Not saying this is happening but only thing I could guess at that would cause this.