No, currently you cannot turn off the sound that the class starts MessageBox. This is a function attached to the system, and if the device is not disconnected, sound will be played.
XNA async , ( ):
Guide.BeginShowMessageBox("Title", "Text", new List<string>() { "OK" }, 0, MessageBoxIcon.None, new AsyncCallback(YourCallback), null);
MessageBoxIcon , ( Windows Phone MessageBox).
: Microsoft.Xna.Framework.GamerServices.