Gunpowder will not fire if it is wet. So technically, you could fire a musket in the rain as long as you were careful about keeping your powder dry. One of the best examples of this is the Battle of Fort Necessity at the beginning of the French and Indian War. Prior to the battle, George Washington (then a Colonial officer leading a small force of Militia and British Regulars) retreated from a superior force composed of French Canadians and Native American allies. They took shelter in 'Fort' Necessity (essentially a circular earthwork with some logs for walls and no ceiling) which was situated in the center of a field surrounded by forest. The resulting battle took place during a huge rainstorm. As they had little shelter against the rain, Washington and his forces were unable to keep their powder dry and were quickly forced to surrender. The French and Native Americans did not have the same issue as they were sheltered somewhat by the trees. As for large scale battles, I don't know if 'postponed' is the right word, but generals would not send forces to fight when they knew their primary weapons would be useless.

There was a discussion here with user/DonaldFDraper about malfunctioning muskets and rain. TL:DR it could be hard to operate a flintlock in the rain.