My water service has been turned off but when it rains I can collect about 2 gallons of water per hour from my faucet. How is this possible?

Beats me..

Your plumbing system is enclosed and pressurized system. There is really no way for rain water to get into your pipes. Maybe other Experts can step in and offer their explanation...

How do you know it is "...rain water ..."?? Do you have a water collection tank on the roof ?

Possible to have a rain water collection with a pressured pump and/or a system that would be normaly used in plant irrigation only that has been back feed into the plumbing.

For it to come out of the sink, the water would have to be collected above, since someone has to force it out.

If the plumbing is done correctly this is not possible

Both Milo and Chuck are correct. With a normal water system this would be impossible. There's some details you're not telling us. Is this some sort of special installation such as Chuck describes? Let us know, Tom