Final answer:
The most likely diagnosis for the affected burn extremity is frostbite.
Step-by-step explanation:
The most likely diagnosis for the affected burn extremity with the symptoms described (tender, numb, white, and cold with barely dopplerable pulses) is Frostbite. When the body is exposed to extreme cold temperatures, such as in frostbite, the arterioles constrict to minimize heat loss, resulting in reduced circulation and a whitish hue in the skin. If the temperature drops too much, the skin can actually freeze, causing frostbite.