At its core, climbing is essentially a functional movement body weight exercise. Every movement you make in during a climb is relying on your muscles to deal with the way your body moves in space, controlling the internal forces to produce efficient and effective strength.
While climbing does promote a total body strength increase, including important areas such as the core and lower back, there is a tendency to favor strength development in certain muscle groups over others. While a degree of total body strength is needed to perform at a high level, the fact of the matter is that the upper back and shoulders do the brunt of the work on any given climb.
This is an issue that I myself have problems with, and the vast majority of my back problems stem from a strength imbalance in my upper body. Climbing favor the upper and mid fiber traps which is the major muscle group the runs from the base of the skull, connects to the scapula, and runs down the spine to T-12. The lower fiber traps have a tendency to be thrown off balance by the overdevelopment of the upper/middle traps which can be seen as the lower corner of the scapula, sticking out from the back. This can result in pain in the mid back, through the area of the lower traps.
Overdevelopment of the back and shoulder also has the effect of forcing the shoulder into a position of internal rotation. Take a look at the resting position of your shoulder in the mirror. If from the side, it appears that your shoulder is shifted forward, there is a good likelihood that the overdevelopment of the shoulder muscles is not being properly offset by the pectorals in the chest. This would once again put strain through the mid back and potentially cause pain or tightness.
Exercises that isolate the lower fiber traps and the chest are an important part of any climbing regime to keep the body balanced. Extreme strength imbalances can cause serious back problems. Resistance exercises that isolate the lower traps, pushups, isometric chest press holds and flexibility/mobility training directed at the shoulders are important to maintain a strength balance in the body.
Your body is a machine, it needs to have all of the parts in the right place to work properly.