Thank you for your question and welcome to Just Answer.
Your dentist did not take enough care to protect your gums and lips during the bleaching procedure.
You have suffered a mild chemical burn and the swelling is purely inflammation as a result of your body's attempt in healing the affected areas. The best thing you can do to accelerate the normal healing process is to take over the counter anti-inflammatory medication such as Advil or Motrin. In addition, you can rinse with warm salt water which will soothe the tissues and also help in healing. The majority of the work in healing will come from your body's natural healing ability so hang in there and you will see improvement soon.
I hope you find this information helpful. If you have any concerns regarding this issue that I have not addressed, please let me know and I will get back to you with more information.
George McKee DDS