There are three problems that typically arise from playing paintball in cold weather: fragile paintballs, inconsistent gas expansion and mechanical paintball gun problems.
Cold paintballs are fragile and have a tendency to break in the barrel. This can be overcome by using paintballs with a thicker shell or use paintballs that are specifically designed for cold weather use.
Inconsistency comes from gas that doesn't properly expand. While compressed air usually performs pretty well at lower temperatures, CO2 is notorious for its inconsistency. Carbon dioxide is in a liquid form inside a tank and must change to gas when it expands. This is an endothermic reaction which means it requires heat form the outside air to turn the liquid into gas. Colder air means a slower rate of expansion which means turns into paintballs that shoot slower and liquid CO2 that stays in the gun without expanding. Effects of this can be minimized by using an expansion chamber, but CO2 can never be guaranteed to be consistent and reliable in cold weather.
Paintball guns are never as reliable in cold weather since they are not primarily designed to work in cold weather. One of the largest problems is that o-rings are designed to work at higher temperatures. If you play in below freezing temperatures your o-rings will become harder and will not as effectively perform their job, often resulting in leaks and inconsistent firing. While there is no true solution to this problem, a well lubricated gun will limit problems.