The fireplace insert must be clear of all objects before it can be removed. If the fireplace is powered by gas, the gas must be turned off before anything else is done. Once the fireplace is empty of all objects, you can slide it out. If powered by gas, pull it out just a little so that you can guide the gas line through the hole it is sticking through. Disconnect and remove the ignition that lights the logs. Remove dust and small debris by using a vacuum.
Pull It Out
If it is a gas fireplace insert, once the lines are out, the process for removing the insert is the same as removing a wood stove insert. Get the help of at least one or two people to remove the insert. Pull the edges until it clears the fireplace. After it is removed, patch any holes or deep scratches that were created from the insert's installation or removal with heat-resistant material.
Put a dust sheet down around the fireplace and all the way to the door through which the insert will be taken. Even though the insert was cleaned out with a vacuum, it's still likely to be dirty. The fireplace itself will need to be sealed off with doors or a damper to prevent heat from rising out the chimney.