I've never milled anything, but I have done a fair amount of bucking pine, and other heavy pitch producing evergreens. Gummed up chains, and bars get sprayed down with a liberal coating of generic WD40 type spray. The pitch that gets on my hands, and saw handles get cleaned off with a shop towel soaked with two cycle mix.
I've never cleaned a chain until a couple weeks ago. smoked & burned on stuff from cutting an oak stump. Put them in a small bucket of simple green for a few days, and prior to sharpening hit them with a wire brush, big stuff came right off. Spritzed with some kroil prior to storing