How to Fix a Lens Cover That's Stuck
Most of the time, your automatic lens cover issue can just be solved by resetting the camera. To do so, turn off the camera completely, and then remove the batteries and memory card. Wait at least ten seconds. Afterwards, reinsert the batteries and memory card, and turn on the camera again.
If that doesn't fix it, then assume that there's some sort of particulate trapped within. Using compressed air or some other air-blowing device, gently (gently!) blow room-temperature air into the cracks around the lens cover to try to dislodge the particle. Try to angle the air such that it won't just blow the particle deeper into the mechanisms, potentially causing even more issues. A low-power vacuum might work better.
If that doesn't work, try sliding a thin strip of strong paper between the shutters of the lens cover and work your way completely around the lens.
Still not working? Try very gently tapping the lens barrel of the camera – this may require turning it on, depending on your camera - tipping it at a variety of angles while doing so to attempt to dislodge the particle. You may have to repeat a few times to get it to work.
A little bit more on the invasive side, but you can try gently pressing against the lens cover, using a piece of paper as a medium, and very gently rotating the shutters in the direction that they are supposed to go. This might allow the guide pins of the lens cover to recatch themselves.
Hmm, still not working? Well, beyond those steps, any possible fixes become quite invasive and require a fair amount of mechanical savvy—and just might void your warranty. Check with your camera company to see if they will cover the cost of an automatic lens cover repair at either free or reduced costs. If not, and you're not willing to pay full price for either the repair or the replacement, you might want to consider doing a little DIY camera repair. Check online for fixes specific to your camera.
For a more detailed guide to repairing stuck automatic lens covers, as well as links to mechanical fixes for specific cameras and a number of very helpful comments, check out this DIY Camera Repair article.