My lips are extremely swollen after hyaluronic acid injections, what do I do? (Photo)

I agree you are having more swelling than average. Lips do tend to swell more than other areas, and in the first 48 hours, a dramatic response like you are having is possible. It will get better, but it may take a couple days. 
Some things that can help are to ice the area, but you must be gentle when doing this so you don't push the filler around and get an asymmetrical result. I also recommend an antihistamine (Benedryl, Zyrtec, etc) which can help with the inflammation. Keeping your lips well moisturized with something basic like Aquaphor will make things more comfortable too. 
Steroid pills (just a few doses) can also be prescribed to help the swelling improve more quickly. The doctor who did the injection should see you first and then decide whether steroids are right for you.
If the swelling is intolerable, or your doctor thinks you are having an allergic reaction, an enzyme to break down the filler can be injected to help things settle down. Your doctor and you can decide if this approach is the best option.
Hopefully things will improve quickly and your end result will be excellent.
Best,
Dr. C-P

Some swelling, tenderness and pain is very normal after fillers are injected into the lips.  However, the amount shown in your picture is above average.  I would recommend following up with the person who injected you as soon as possible.  You may be having a (rare) allergic reaction to the filler.Best of luck to you.