Heat 1 tablespoon oil in a medium pot over medium heat. Cook onion until soft, 8 to 10 minutes. Add garlic and tomato paste. Cook until fragrant, 2 to 3 minutes.
Drain beans, and add to pot. Add tomatoes and water. Bring to a boil. Reduce heat, and just barely simmer, partially covered, until beans are tender, about 3 hours.
Meanwhile, prepare an ice-water bath. Bring a medium pot of water to a boil. Blanch dandelion greens until tender and no longer bitter, about 9 minutes. Transfer to ice-water bath using a slotted spoon.
Once beans are tender, add vinegar and 1 teaspoon salt. Drain dandelion greens, and add to pot. Rewarm if needed.
Divide beans among 4 bowls, and top with feta and dill. Drizzle each with 1 teaspoon oil.