It is possible for an expansion tank to introduce air into the hot water system if the internal diaphragm leaks but in that case, within a few days the air chamber in the expansion tank would become saturated and you would see water leaking from the Temperature & Pressure Relief Valve.

The original post states that this problem occurs every time they use the hot water, but not when they use the cold water. If the water heater is installed to code specification that is a physical impossibility. The plumbing code states that the only valve permitted between the hot and cold water supply is the "cold water shutoff valve" on the cold water input line to the water heater. When the cold water supply valve is open there is a direct connection between the hot & cold water systems, and the pressure in both hot & cold water distribution piping remains equal.

In a standard municipal supply system, when thermal expansion occurs in the water heater a volume of water equal to the expansion back flows into the cold water line and ultimately out through the house water supply line to the municipal main, however even when that thermal expansion occurs the pressure remains equal in both the hot and cold water distribution systems.

When the municipal static head pressure to the structure exceeds 80psi we are required to install a PRV (pressure reducing valve) on the cold water supply line. Most PRV's have check valve feature built in so we are also required to install an expansion tank on the cold water supply line to the water heater, but here again, even thought the expansion tank absorbs the thermal expansion as the water is heated, none-the-less, the pressure on both the hot and cold water systems remains equal.

Home well pumps also have a built-in check valve, therefore we are required to install an expansion tank on the cold water supply line in the close proximity of the water heater, however, here again, the pressure remains constant on both the hot and cold water systems.

This then leaves us with two possibilities.

1. Someone foolishly installed a check valve on the water heater supply line.
2.The house is supplied by a well pump and the water from the well has an excessive amount of entrained air, which is being released by the heat. Generally this indicates either a problem with the water or the pump is not deep enough in the well.

Andrelaplume2 stated that they had air sputtering out of the faucet this morning, but it was an isolated case and not representative of normal operation. They went on to say that perhaps the water supply company was up to something overnight. You are correct, it is quite possible that they had to open the main to make a repair and in the process air got into the line, In that case the problem will correct itself in a very short time.

A vacuum relief valve is designed to let air into the system to prevent several forms of damage. Do a Google search for "Vacuum Relief Valve" and you will find plenty of info on why they are installed and what damages they prevent.

If you have a bladder type pressure tank, the air chamber may be leaking...check the air pressure in the tank. Also, if your check valve is leaking and a bit of the water drains back to the well, air can be introduced via the vac. rel. valve or any leaking faucets.

If there is excess air dissolved in the water heating it will force it out of solution.

Save your back. You don't need a shovel to remove a pitless adapter.