Disable or Mute Trash Sound
The sound file for moving an item to the trash is located here:

Go to your HD > System > Library > Components > CoreAudio.component (right click and choose "Show Package Contents") > Contents > SharedSupport > SystemSounds > dock > "drag to trash.aif"

This "drag to trash.aif" is the sound file responsible for the audio effect.

Back up the current file just incase and replace it with a silent one (a quick google search will do - you might need to get the silent file as mp3 and then convert it to aif).

You may need to restart for it to take effect.