Question: I have seen some general comments online about running a 6V DC motor on 9V Battery being OK. 
I am looking to run an air mattress pump originally designed for 4x 1.5V D batteries (6V DC) from a universal car laptop adapter. The lowest output voltage on the car adapter is 9V.

Nothing on the air pump motor mentions input power specs specifically. The only markings I can see are the following 3 lines and I get no hits googling each string.
hb-138-101 
rs-380ph-6016rb 
630301

The alternative would be to use an AC inverter and an DC 6v 2A adapter. It sounds like overkill to go from 12v to 120v AC just to get 6VDC, but so is running a car engine. 
Will I burn the motor if I ran it on 9V DC?

Answer: 
Short answer, yes. Both the circuits have same amount of electrons transferred in the same period of time.

**Theory behind it:**

Current is the rate of flow of charge and voltage is the amount of work done in moving that charge from some point to another point.

Now since both the circuits are drawing 100mA of current that means both the circuits have same no of electrons flowing at any particular point. But different amount of work needs to be done to achieve that same flow and hence different amount of power in both the circuits (P=VI)

A good analogy is a water, so current (rate of flow) is a flow of water where as the voltage is analogous to pressure in the system that will cause the water to flow (ie current).

So both the systems have the same amount of water flowing in given time period but one system needs to work more than the other to achieve the same rate of flow.

That means 5V circuit is putting more amount of work to maintain 100mA in circuit compared to 3.3V circuit.
Practically, I have two situations where that amperage is the same. My Electric Vehicle charges at 16 Amps with either a 110 Volt charger or a 220 Volt charger. The 110 Volt charger takes twice the time to charge the car battery at the same amperage. When I look at my electric company's energy use in kWh the total cost to charge my car battery is approximately the same. Hence with 1/2 of the voltage, only 1/2 of the power is delivered per hour.

Back to the water analogy. Is it the case that with the 220 Volt charger, my "water flow" is the same per hour, but with the 220 Volt charger I am using "two 110 Volt hoses" (hence 2x the amount of power per hour compared to a 110 Volt charger)??