My gum is sore where my implant screw fell out, can it be infected?

if the screw over the implant fell out , then the gum will grow in to the head of the implant and try to cover it bacteria could also grow in or around the implant keep the site clean. See your dentist as soon as possible to place another healing screw before you decelop infection and bone loss 

The failure rate for dental implants is around 3% to 5%. This means that for every 100 implants that a dentist puts in, about 3 to 5 will fail, and have to be removed (or fall out on their own). There are 3 more frequently main reasons :Infection.Not enough bone to anchor the implant in the first place, resulting in "micromovement".Poor blood supply to the bone around the implant.So i recommended you to talk to your doctor and evaluate your situacion. 

This may or may not be an issue depending on whether food got under the gum covering the implant and has set up a low grade inflammatory reaction. I would see your surgeon to asses and put the cover screw back ASAP.

This happens sometimes and when it does the gums have a way of growing back and covering the implant.  This should not hurt, unless something is stuck in there or something irritated the area.  Keep the area clean and go see you doctor to evaluate.