What happens if a type 2 diabetic with gangrene refuses amputation?

Wet or dry. If it is wet gangrene (infected) it can cause sepsis and death. If it is dry gangrene, it can be monitored closely and you can live with it as long as it doesn't convert to wet gangrene. Gangrene is dead tissue, bacteria loves dead tissue- so dry gangrene is a nidus for infection. Talk with your doctors to help weigh the specific risks, benefits, consequences and overall prognosis.
Bad prognosis. Doctors are careful to recommend amputation only when medically necessary. Gangrene can spread to the rest of the body and cause life threatening illness.
Many... Unfortunately if there is gangrene and infection the problem is likely to spread higher in the leg. If left untreated the infection can go in the bloodstream. This is very risky and the person may die.
They usually die. Usual result is septic shock and death if the damaged tissues are not removed ahead of the gangrene process.