Actually, sktn, they're not talking about rain, which the heat exchanger is protected from by a weather shield. They're talking about using a heater to take water from say 40 degrees and heating it to under, or maintaining at 68 degrees.
The coldness of the water going through the heat exchaner pipes versus the flame is what can cause the condensation they're talking about. Combined with extremely cold makeup air being drawn into the system, this create corrosive condensate and cause sooting which further exacerbates condensate because now your partly blocking the venting of the combustion gasses, which instead condense into a sticky greenish substance that's difficult to clean.
In other words, if you were going to heat 40 degree water to 80, by that time the exchanger would be running warmer due to the 80 degree water running through it. That's why its not an issue to start up your heater in the spring even with cold water...in a set number of hours your water will be warmed past the problem area.
At least, that's my layperson's understanding after discussing the phenom with a mfg rep.