You very well may have a case, but it will depend on a number of factors.
First, you will need to determine whether the release covered injuries other than from UV lights. To do this, you'll need a copy of the signed release and will likely want an attorney to review it.
Second, you will need medical confirmation that your burn was indeed caused by chemicals. That may be difficult to prove, and it may be even more difficult to prove that the chemicals came from the tanning bed. Then again, there may proof that a certain cleaning agent the salon used is known to cause burns.
These factors will all go into determining whether the salon was negligent in its actions towards you, the customer.
Your best bet is to consult a personal injury attorney. Personal injury attorneys normally do not charge for consultations and will be able to look into the claim in further depth.
My response does not constitute legal advice but is provided for the purposes of advertising and education. Please contact a local attorney if you wish to obtain confidential legal counsel.
I agree that you need to consult a personal injury lawyer immediately. You don't say if you have had medical treatment or whether your doctor can confirm that this is a chemical burn. If so, you claim is much better. However, the release will have to be examined by a lawyer to find out if this is covered. The release will be viewed in the light most favorable to you by the courts since they created the language in it.