Well for one thing, as the air becomes more humid it becomes less dense. IOW, density altitude increases which will affect performance and aerodynamic lift. But this is negligible compared with the downpour that rain itself falls from or out of. It isn't necessarily the same thing as destroying lift per se, but rather creates a downward force which if one is attempting to hold altitude may cause an increase in AOA which would would slow the plane down slightly due to the induced drag created. Most light rain won't be much of an issue for the average transport aircraft, unless you're talking thunderstorm precipitation.
An increase in airspeed might be required due to the downpour which will inevitably set up some windshear where that may cause a decrease in airspeed, necessitating an increase in power to compensate for the loss of airspeed.
