I've used olive oil and canola (rape seed) oil on natural furs like beaver, wolverine, bear, martin and some hairs such as deer to clean off sticky substances. They are just natural oils, so no harm. Use a very small quantity and gently stroke it in, then carefully wipe it away with tepid water. Whatever residue remains is not harmful to the fur. A few applications may be necessary in severe cases. Soap is a killer for natural fur, of course. I suppose natural oils would work well on artificial furs as well, but perhaps try them on a wasteable scrap. I wouldn't want to use mayo on natural fur for the other ingredients in commercial products, but a good luke-warm wash afterwards may get rid of those, too.