How do I plug a leak in my refrigerator's water fountain casing?
The root cause may be the closure flap in the ice hopper delivery chute. There is a spring-loaded flap up there which opens when you actuate the dispensing lever. Make sure it is closing all the way and sealing (there is a foam seal on the "ice" side of the flap, check it for damage) or the ice in the bottom of the hopper can melt and drip down into the tray. Also ice sometimes gets trapped in the chute and does not fall all the way down into your container, it then melts and drips into the tray.

Shine a flashlight up into the chute and check it out.