Later this year I will be buying a house. Since my elderly grandfather
will be moving into the house with me, he is providing the down
payment, about $160,000. From what I understand, the money is coming
from an investment account of his and from a revocable family trust.

It's not really logistically feasible for my grandfather to cosign on
the mortgage with us, and we wouldn't consider it an advance on his
rent for living with us since I believe that's rent income I'd have to
declare. What are the tax implications of taking the money as an
advance on my inheritance?

Lucky you! 

Basically, you owe no taxes on gifts received. The giver pays taxes on
amounts over a threshold. See other posts in Google answers for
details on these threshholds.  But they is an exception for family
members such that your blood relative can give you upto $1 million
during their lifetime without triggering any tax for themselves. 
Again, check other Google Answers on this, or search IRS for "gift
tax."