Did you know mangoes belong to the same plant family as poison ivy and that the skin of a mango can give you that same great contact dermatitis as if you played with poison ivy, poison oak, or poison sumac? I did not know that. That is a factoid that (thankfully) I did not learn as a result of personal experience. I was seeking the chemical structure for urushiol to add it to my indirect poison ivy blog post and I read that if you have contact dermatitis from poison ivy or one of the other urushiol-containing plants, exposure to the cut skin of a mango can be a highly unpleasant experience. From there, I found a nice list of Contact-Poisonous Plants of the World that you can peruse for education and entertainment.
Obviously people eat mangoes all the time. The edible portion isn't going to cause you problems. However, the vine of a mango contains sufficient urushiol to cause a reaction that rivals or exceeds that from poison ivy. The skin of the mango contains enough urushiol that if you are already sensitized to it, you will probably get contact dermatitis from exposure, usually on your hands, since most people don't bite into mangoes.