https://support.panasonic.co.nz/.../DMREH57%20Operating%20Instructions.pdf
Above is your user manual and below is your service and repair manual.
http://www.eserviceinfo.com/downloadsm/33957/PANASONIC_DMR-EH675.html
You unit's draw should open by using either the key on the actual unit, or the remote, both should cause it to open. When these units do not open, it is only a few things it can be. A: The "Load" motor has Failed, The "Drive" to the "Load" motor has failed, or The Microprocessor, isn't functioning correctly, or a combination of any or all.
If you work through the troubleshooting in the manuals, you maybe able to effect a repair.
Now if you feel that it you do not want to do it. It is my professional advice is to get you set looked at by a Panasonic, Authorised Service Centre, and, when there, ask for free "Quote", this way, you will be able to make an informed judgment, as to repair or replace. As these days seemingly insignificant SYMPTOMS, often cost as much to repair as it may cost to buy.
Sometimes you can open the draw to get out an disc within, by using a straightened piece of paper clip, now look for a small "Hole" in the front of the tray, push the wire into this hole, and keep pushing in until you hear/feel a "Click" keep push in, gently, and the draw will.should "Pop" open.