here is the troubleshooting procedures from lexmark.
If it doesn't help, then the problem might be either faulty cover sensor (which is just a simple switch that is pressed when the cover is closed), displaced sensor (it might have come off from the chassis) or that the extrusion (or pin) on the cover worn off / broken and not pressing the switch anymore.
Sending your printer for a repair is probably not economically viable. You can attempt fixing it yourself, but you probably won't get the full functionality back, plus such a home fix would probably bypass printer's safety feature (it won't know anymore if the cover is open or closed, you will have to "tell" it to the printer).
plus, the price for a new printer is probably around 50$, and the solution might take you an hour or two if you have no technical background. In case you opt for a new printer, recycle your old one responsibly or donate it to local tinkershop for parts.
Disclaimer. Do not attempt the solution if you do not take full responsibility for whatever the result of you actions will be, either directly or indirectly, etc. etc. I take no responsibility for loss of life, sanity, pets, Apocalypse and so on.
*1Disconnect the power from the printer. Locate the cover sensor switch, it is probably near the back of the printer where the paper is loaded. If the sensor is in place, check if it is ok by pressing it with a non-conductive object, such as a plastic pin, if the sensor was knocked off, procede to *4.
*2 Make sure you are clear of any moving parts of the printer and that carriage path is not obstructed by anything. Power on the printer. If the printer powers on normally and doesn't display the "cover open" message, sensor is ok, if it doesn't (you should try a couple of times, the sensor probably produces a "click" -sound when it is pressed), the sensor is faulty, proceed to *5.
*3Power off the printer. Check if the "pin" or extrusion on the cover is damaged. If it is, you can either try to glue a piece of plastic to extend it, so that it engages the sensor firmly when the cover is closed, or you can cut a piece of plastic to reside on the sensor att all times (thus you will have to power down the printer, remove the piece and then power on the printer in order to get the carriage into central position to change the cartidges).
*4 Locate the sensor, if you have clear access to it, press the sensor with a non-conductive object, you should hear a "click" sound. Make sure you are clear off the moving parts of the printer and the path of the carriage is not obstructed with anything. Power on the printer. If it powers on wihtout "cover open" message, the sensor is ok. Power off the printer. You can either try to put the sensor back in place (might be problematic as you have to poistion it correctly in order that the cover can press on it when it is clsoed) or you can try to put the sensor outside of the printer and operate it manually, by putting a piece of non-conductive amterial inot it when the cover is closed and removing the piece when you need to cahnge the cartridges. If the sensor is faulty, proceed to *5.
*5. If you have soldering skills, you can replace the sensor with the same make and model, or you can replace it with another switch and put it ouside the printer to operate it manually, i.e. put it into "off" position when the cover is open end "on" when the cover is closed.
If you have any questions, you can leave your message here with pictures.