I definitely think that you could say, with a somewhat humorous meaning, that you "feasted on nuts". It would mean that you ate a lot of them, so many that it equaled a huge meal.
I don't think you would ever say you "dined on nuts", as "dine" implies that you are eating a meal, and I don't think anyone would make a meal of nuts.
"Snacked on nuts" is definitely the most correct option, as nuts are a snack food.
"Feasted" might also give an indication of the person's state of mind. "After 5 days staggering across the desert with nothing to eat but half a mouldy biscuit, he came to a small oasis where he feasted on nuts." (I know, I know, it was probably dates, but you get my drift).