All refineries have to shut down to switch from winter to summer gasoline blends in the spring and back again in the fall.
I don't believe this is correct. They may need to make certain alterations in processing and in certain additives, however, refineries typically run for at least a year, often 2-3 years whenever possible. Larger refineries, with multiple trains, can even run continuously, shutting down portions of the refinery for maintenance while other portions continue to operate (outside of catastrophic failure, obviously).
Summer gasoline costs more because the inexpensive additives that are used in the winter cannot be added in the summer, otherwise the gasoline would have too high a vapour pressure for the higher summer temperatures and cause vapour lock. Certain (light, inexpensive) additives are used in the winter to raise vapour pressure to make for easier starting in cold weather, hence lower cost gasoline.
Of course, gasoline also goes up in price in summer due to higher demand as people drive more. Production of heating oils, kerosene (for jets), diesel, and asphalt (for road construction), depending on the refinery product slate, also play into the pricing of all the various products from a refinery. More on refineries in the link below: