When I went to a gym to pick up a membership form, they gave me a parental consent form as well.
I understand that means there is some risk attached to going to a gym for children. Aside from pulling muscles, what other health risks are there?
Having little Timmy at age 3 doing curls is not the same as having big Bobby at age 17 doing curls, so for the purpose of this answer I will assume the base age of 6 as an appropriate minimum for being defined as a youth / child.
As far as health risks go, they are really no different than that of any other sport or physical activity; it all boils down to using proper form in a safe and supervised environment. By supervision I am not referring to some nanny making sure they aren't rough-housing, but rather an experienced and knowledgeable personal trainer who can ensure they use proper form, spot the weights, and build a safe and age appropriate training program for the child.
Keep in mind that there are also psychological benefits and risks associated with sports and resistance training. While the activity may be very beneficial for the child physically, if they treat it with the wrong mentality then it can be dangerous to them psychologically (think sports breeding hyper-competitiveness). By introducing weight lifting at an early age you risk the child becoming hyper-narcissistic (among other things) if you approach it with the wrong mentality.