Does a doctor need to see this sore on the big toe?

It looks like it might be infected. If it gets worse and progresses, then you will need to see a doctor. Don't let it go on too long, or you could lose your foot. 

If it starts turning brown/black, and is unusually painful, then it's an emergency. 

See a podiatrist

No one is correct completely. The red bump is a pyoderma granuloma which is a type of over growth tissue from repeated irritation. In this case the nail. Repeated cleaning will only make it worse. The nail needs to have the spicule removed and the pyoderma cauterized with a silver nitrate stick. Podiatry or some UCs

Matt is right. That's an ingrown toenail which is now infected. But given it's severity and the discoloration along your nailbed, you should see a doctor for some antibiotics. You're beyond home remedies.

peripheral neuropathy causing gangrene to form .

if youre not sure i would go see one

Soak this with peroxide for a bit then rinse well apply neosporin or some antibicots and apply a gauze cover tape up let on every night, shower the next day, repeat this should heal quite nicely if no infection occurs.

looks like an ingrown toenail that has gotten infected, soak your feet in a tub of hot water with apple cider vinegar, this will help with the infection, apple cider vinegar has natural antibacterial qualities, it will also soften up your nail and help it move out of the nailbed, then you should dry it and apply Neosporin and clean gauze, do this till it is healed, you should let your toenail grow out and don't cut it so short, toenails should be cut strait, not rounded like fingernails, the edges of your toenail should be outside of your nailbed so that it doesn't become ingrown and become infected