Removing app icons from the Dock has been the same since the very beginning of Mac OS X: grab an icon and drag it off the Dock into a poof of dust, whereby the icon is no longer displayed in the Dock. That behavior has changed slightly with OS X Mountain Lion, presumably to prevent accidental deletion of Dock apps. There are still two easy ways to remove Dock icons in OS X 10.8 though, pick either approach that works best for you.
2 Ways to Remove Dock Icons in OS X Mountain Lion
Click, drag away from the Dock by about 5cm, and hold for a second or two until the “poof” icon appears and then release.
Drag and drop the Dock icon into the Trash
For the impatient, the drag to Trash method is the quickest and it’s demonstrated in the video embed below. It works for any icon in the Dock, be it for an app, folder, shortcut, or otherwise. You can not remove active applications from the Dock, however.
This may seem like an elementary tip to cover, but any time longstanding things change it can confuse even advanced Mac users.
This behavior will likely continue from OS X 10.8 onward. If you’re encountering any problems with this, try dragging the icon further away from the Dock and holding it a second longer.