Greetings, I am Dr. Pat. I have worked with birds for many years. I will do my best to help you.
Two things to try, and both are messy and will be stressful, but you have to try.
Olive oil may unstick but not remove the material, then you will have to wash that off. Make sure he has eaten and is hydrated, they become exhausted with the trapping. Have some sugar water ready to give him. Set up a recovery unit before you start. Go to this link
for some ideas.
Use a VERY TINY amount of oil and rub it into the goo. Wash him with very warm water (105F) with Dawn dish detergent then rinse thoroughly. Warm him up immediately. Use sugar water if he is exhausted.
The second method is a little more dangerous but works much better.There is a product called "Goo Gone" available at hardware stores and most Walgreens-type stores. A very very tiny amount will work, then you will have to do the bathing procedure as outlined above. Goo Gone is a little more toxic but will work better, you just have to work fast and bathe it off as soon as it takes off the sticky trap.
Move the bird to an aquarium, box or carrier with soft towels in the bottom, no perch, and food and water in low bowls that can be reached easily. Put the whole thing on a heating pad on low or medium. Keep the unit partially covered, warm and quiet. Do not try to force food or water. You can offer warm cooked rice, pancakes, cornbread, grapes, melon, greens in addition to normal food.