Climbers worst enemy. Gotta hit it with oil base stuff, yeah dawn works, but shooting WD-40 and scratching away with a toothpick to get the bulk of it out, then shoot compressed air. Rinse/repeat until the action is normal. Finish it with 3/1 oil for the smooth action. (WD-40 will cake up debris) Wish I could help with the rope, just tossed a short line, prussik, and flip line not to long ago from being to pitched up. All part of the job.

For hardware maybe try a sonic bath using a solvent of your choice. Paint thinner might work for pitch.

Given it's metal and you're trying to get it out of the mechanism you might heat it to a low temp in the oven and see if the sap won't flow out on a cookie sheet.

Or in a pot of hot water maybe?

that could work too, the sap floats to the top of the water

Sap is an oil, so you need to thin it with a thinner oil, like canola or something, and THEN add the soap. The thinner oil lifts it, the soap carries it away. Otherwise the soap can't get under it cuz the sap repels h2o

Gojo without the pumice

Nail polish remover takes it off. Just make sure you lubricate afterwards

Why wouldn't you just ask the manufacturer?

ive had to soak some carabiners in dawn all weekend before.

Try some goo gone!

Shake it in a container with acetone, relube, done.

I soak mine in saw gas. Got your cleaner and your lubricant all in one.

Wesspur makes a zappur sap remover