Most people don't climb long enough to wear out their back muscles. But they do get a workout when climbing steep roads, as the lower back muscles are used when climbing.
If you haven't experienced this, go out and do a 2-hour climb seated. Your back will be tight.
I make a point of standing every few minutes on a climb, or whenever I feel my back start to tighten up. It helps immensely.
Managing Director, Undiscovered Country Tours
If you have a weak core then any strenuous cycling can lead to back pain. I only stand to get on top of a gear or sprint when climbing, otherwise I'm always in the saddle.
Perhaps I'm missing something, as I'm a relatively new cyclist, but not new to exercise and muscle roles during exercise.