I have a tall (~60 feet) fir tree next to my house. It has grown to the
point where the branches are starting to touch the side of the house, even
though the base of the tree is ~10 feet from the house. What is the best
way
to trim the branches - remove them at the trunk, or just cut them enough
so
they are no longer near the house? If I start removing branches that grow
enough to touch the house, eventually I will have chopped off most of the
branches on that side of the tree. I'm thinking I should just cut them
off
so they are not too close to the house.

Get rid of that tree, you're looking for trouble.

I live in Oregon. Tree central. The bleading heart activists would firebomb
my house if I cut it down :-(

On a more serious note - I live in a small town in the Wilamatte Valley,
Oregon (Lebanon, about 40 miles SSE of Salem, which is about 50 miles south
of Portland) where the city enacts draconian rules and regulations. They
won't let me remove it, and it's not worth the legal fight it would take to
get permission. Out of curiosity...what kind of trouble could this cause?