Can you get frostbite from eating popsicles?

No: If you held ice against your skin for a long period of time, it is possible to cause some local tissue damage to the area- but very difficult because you would have severe pain in that spot first and therefore unlikely to hold i've in one place long enough to cause a problem to your own tissue.

Unlikely: At or below 32 °f, blood vessels close to the skin constrict. Prolonged lack of blood eventually leads to freezing and death of skin. "frostnip" or first degree frostbite only affects the surface of the skin. If freezing continues, second degree frostbite can occur. However, it is unlikely that one would continue holding a popsicle long enough for any real damage to occur. (one would hope!).