The most common verb for this in AmEng is eat. There are other areas in the English-speaking world that use drink. I think that drink would be inappropriate if the soup is a hearty vegetable or chowder, however.
There is the legendary retort by Sir Winston Churchill to Lady Astor [ the two didn't get along at all! ] when she said that if he were her husband she would poison his soup: Lady Astor, if you were my wife, I would drink it.