If this code only applies to the downstream O2 sensor then it will not affect the open/closed loop operation of the engine. The downstream O2 sensor controls the heat of the catalytic converter. It slightly richens the mixture to make the cat heat up faster, and it leans it a bit if the cat gets too hot. Basically it keeps the cat in the optimum temp range to work right. It is true that it could damage the cat by making it get too hot if it’s not sensing correctly. BUT, the rear O2 sensor will not cause the car to remain stuck in open loop. The front sensor does that.
Presonally, I’d fix it. It can in fact damage the cat which would certainly cost more than $300 to fix. And while helokit is correct that it might not be the sensor, the sensor would be the most common failure of this type. The heating elements in rear heated O2 sensors do go out pretty frequently.