For myself I found that correcting my posture had a positive effect on reaction time. Specifically when I play counter strike, if I sit upright with correct posture, I can react better.
My theory is that bad posture leads to neural tension. The nerves are literally being stretched out in ways they were not designed. The nerves then act in a protective manner to prevent injury, which gives a constant level of muscle tension which is not optimal.
If you correct your posture, the nerves will have some extra slack, and in turn they will let the musculature relax and perform optimally.