Reduced risk of solar flares: In 1859, a massive solar flare and geomagnetic storm hit the Earth.[1] Magnetic storms induce electric currents in wires. Unfortunately for us, by 1859 we had wrapped the Earth in telegraph wires. The storm caused powerful currents in those wires, knocking out communications and in some cases causing telegraph equipment to catch fire.[2]
Since 1859, we've wrapped the Earth in a lot more wires. If the 1859 storm hit us today, the Department of Homeland Security estimates the economic damage to the US alone would be several trillion dollars[3]—more than every hurricane which has ever hit the US combined.[4] If the Sun went out, this threat would be eliminated.
Improved satellite service: When a communications satellite passes in front of the Sun, the Sun can drown out the satellite's radio signal, causing an interruption in service.[5] Deactivating the Sun would solve this problem.
Better astronomy: Without the Sun, ground-based observatories would be able to operate around the clock. The cooler air would create less atmospheric noise, which would reduce the load on adaptive optics systems and allow for sharper images.
Stable dust: Without sunlight, there would be no Poynting–Robertson drag, which means we would finally be able to place dust into a stable orbit around the Sun without the orbits decaying. I’m not sure whether anyone wants to do that, but you never know.
Reduced infrastructure costs: The Department of Transportation estimates that it would cost $20 billion per year over the next 20 years to repair and maintain all US bridges.[6] Most US bridges are over water; without the Sun, we could save money by simply driving on a strip of asphalt laid across the ice.
*Mostly