Yes, they do. Tomatoes are a climacteric fruit (tomatoes are berries, stupid).
Climacteric fruit produce ethylene, which triggers and promotes ripening, a complex process that softens and–more importantly—sweetens.
Others are apples, bananas, pears, apricots, peaches, plums, avocados, nectarines, and blueberries.
Citruses, raspberries, strawberries, and cherries, grapes, pineapples, melons, and pomegranates are non-climacteric.
Fry green tomatoes, but you can ripen half-green tomatoes in a sunny window with the stem end down.