I live in an investment property of a relative. He is facing foreclosure and the bank is trying to serve him.

He does not live here and is out of town on his primary job 75% of the time.

I have not accepted any legal documents from the people ringing on the door bell because A. I am not the legal mortgagee, and B. I barely see or speak to the person.

Does it buy me more time by not accepting these papers and waiting for them to serve him elsewhere?

Perhaps some. I'd assume your time to vacate depends on when the sheriff's sale happens, and that won't happen until the owner has been served notice (depending on your state's laws) .
There is generally some form of alternate service that will eventually happen, perhaps by publication or mail to last known address, so I'd assume the extra time this creates won't be significant.

Sorry to hear about the situation you are in. I am new to the forum but I am a Realtor with much experience. You may want to read this following link. I would just assume you do have a lease agreement? The following link gives the tenants rights to stay in the unit until lease is over as well as an additional 90 days to find another place if I remember correctly.

http://news.yahoo.com/s/ap/20090522/ap_on_...losures_rentals