A stuck automatic lens cover is a fairly common problem, but it is normally easy to fix. A single grain of sand jamming the cover mechanism is normally the culprit, and you want to try to dislodge it: 1) Try blowing lots of compressed air around the lens cover to clean the mechanism. Or use a hair dryer for a few seconds (don't want to heat up the camera). Use no heat mode if possible. 2) Slide a thin strip of paper between the lens cover shutters and the outer support ring. Work it around 360 degres to clear out any debris that may be in there. 3) With the lens barrel extended and pointed downward, tap the lens barrel with a pencil while you extend and retract the lens by powering the camera on and off. Repeat the above three steps several times. If repeated tries don't work (keeping in mind that it usually does), it then may be necessary to open the lens barrel to access the lens cover mechanism. But before attempting that, first read through the user's comments below for additional non-invasive tips that may work for your particular situation. Especially pay attention to Jeff's tip, several people have had success with it.
The lens cover mechanism is usually accessible for many cameras without needing to open up the camera case. Before starting, a reminder that I hold no responsibility should you damage your camera or worse. These repairs are intended as last resorts on expired warranty cameras that would otherwise be tossed. In the example outlined in the following link, we'll fix an inoperative lens cover on an old Canon A400:
And here's two excellent tutorials for those with slightly more complex four leave lens covers. A Canon A530 and a Canon A1000 are used as examples (hope you don't mind me including the links xscrewdriver).
And finally, found this fix that was particular for the Panasonic Lumix DMC-TZ3 camera. It's simple enough and worth giving a quick try even if your camera is not a Panasonic: