When the weather is warm, my garage door opens okay. When it is cold, it does not. This is also true when it is wet. I understand why the door being wet would make it heavier and more of a load for the springs. But why would it being cold outside make the springs less springy? Or does cold make the door heavier? That doesn't seem right.

I presume your garage door is made of wood, since you said it would get heavy when wet.

The degree to which wood absorbs moisture from the air depends not on the absolute humidity, but on the relative humidity—that is, the amount of moisture relative to the maximum amount of moisture the air can carry. Warm air can carry a larger amount of moisture than cold air. The vapor pressure of water is higher at higher temperatures.

So if there’s a given amount of moisture in the air, and the temperature drops, the vapor pressure of the water is lower and the relative humidity has gone up although the absolute humidity didn’t change. The wood will therefore absorb more moisture from the air when the weather is cold.

I think this is making your door heavier. In principle swings of relative humidity could change the weight of the wood by 10% or more, so the weight of a 300-pound door could be increased by 30 pounds or more.

I don’t think the springs are the problem. Between a cold day and a hot day, the stiffness of the springs shouldn't vary by more than a percent. Furthermore, the effect goes in the other direction from your problem—the springs should be slightly less springy on a hot day.