To sup is to eat slowly, by spoonfuls. When you eat soup, you probably sup it.
You can really only sup soft foods and liquids, like oatmeal and milkshakes and miso soup. A sick person taking gingerly spoons of Jello into his mouth can be said to sup, and so can a hesitant baby eating tiny bites of pureed carrots. Sup essentially means "sip," and its root is the Old English sūpan, "to sip or swallow."