Ungrounded outlet does damage to appliance?

An ungrounded outlet NEVER damages an appliance, even indirectly. The ground protects YOU against electrical shock if the appliance shorts out. A damaged appliance makes the ground necessary, not the other way around.

Damaged Appliance Outlet

Electrically induced damage is very difficult, if not impossible to prove. This is because of the number of channels available for the damage to occur. The answer also depends upon the measures taken to protect the device from stray and spike voltages, by the manufacturer.

In a properly functioning, 2-wire electrical system, the neutral line should never vary from ground voltage by more than a volt. Of course, there are many factors that can affect this. If the home's wiring system is defective (poor neutral connection), this can float neutral 10 volts or more above/below ground voltage. This voltage could disrupt and/or damage sensitive, CMOS control circuitry in appliances.

Someone touching the ungrounded, metal case of an appliance can ground the appliance, sending a spike into the control electronics.

Since the home electrical system is primarily 2-wire (hot and neutral), under stable conditions the lack of a ground would not affect the appliance. The wild card comes in when a spike comes into the appliance's internal electrical system either through the AC line or through the metal case. It's very difficult, if not impossible to determine which was the cause of failure, after the fact.

Of course, consumer electronics (computers, and audio and video equipment) are generally isolated electrically, and only require a 2-wire connection to 120 volt, AC power. Desktop computers are generally 3-wire, so the lack of a ground will affect the susceptibility of these devices to spikes and possible (software/hardware) damage.

Jim, what you are asking is very difficult to prove and/or document. I recommend you contact UL (Underwriters Laboratories) and NIST (National Institute for Standards Technology, formerly NBS (National Bureau of Standards)), Boulder, Colorado and pose your question to them.

You asked a very good question. I just cannot give you a definitive answer. Hopefully what I've given you will help.

Generally, no, but you did not name a specific appliance.

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Ungrounded outlets may be a shock hazard to a user of the appliance.

The auxiliary ground is to protect lives and also the appliance on the event of electrical surge. Yes...! a lack of auxiliary ground can screw the appliance...!