Does the Console contain a warning like "Can not play a disabled audio source"?
The continue button gets deactivated when it's clicked. Try moving the audio source and the script to your Dialogue Panel GameObject. Then hook up the continue button's OnClick() to call the same method, but on the Dialogue Panel.
I moved both the audio source and the script to the Dialogue Panel GameObject. Then I hooked it up to the continue buttons OnClick() event. However, the sound still does not play.
Just as a sanity check, I made it so I stopped disabling my Continue button and it still would not play. Very strangel...
I noticed that when I click the button with my actual mouse, OnClick() is called and the sound effect plays. However, I have a "UI Button Key Trigger" on my Continue button and I press a key to Continue.
I'm guessing that using UI Button Key Trigger to hit Continue with a key doesn't call OnClick(). Is there a workaround for this?