I have a chin implant with wings. I went running. Have I damaged it or is it just swelling? (Photo)

Generally speaking, I have my patients wait 3-4 weeks following facial plastic surgery before doing any form of exercise. Your swelling should be reversible, but may take a while to go back down. Putting cold compresses on the area initially probably would have helped. Oral steroids may be of use now. I would return to your surgeon to have him check it out and prescribe the best course of treatment now.

It's not usual for swelling to get worse after surgery. Exercise at this many days out would not cause that much swelling. My main concern would be an infection - especially if you had an implant other than silicone, I would be particularly diligent about following up for a possible early infection. 

I do not believe you have damaged anything by running 10 days after placement of your chin implant, but the swelling you have 2 weeks after your procedure could be of concern.  It is possible that the running may have influenced the swelling, but normally this should go back down in time.  It is also possible the swelling could be due to something else, so it would be best to contact your surgeon so that you can be evaluated in person.