Much of the answer may apply to many models, but to be precise mine is a Mitsubishi HS-M55(NZ).
I believe the only fault is stiff grease, as it has done this before and it worked fine until the last tape was inserted. :-) That is probably always the case, but what I mean is the VCR was not exhibiting any faults prior to being unable to eject the cassette.
Eventually with repeated trying, it would try and then power off.

But the question is, what can I clean the rails and slides etc with and what sort of lubricant should I put on them afterwards?

I have been searching for a free service manual to no avail, as my user manual has absolutely no specs or mechanical details. Help there appreciated too.

Jeremy, September 2014

Thank you. I cleaned with meths and was really making progress as the mechanisms started to slide.
Unfortunately I taxed the motor too soon and although it ran for much longer without shutting off, unfortunately the smoke got out and it stopped working!
As no parts available now, that’s the end of the road for this VCR. The Sharp has done the same thing - grease gone hard, but needs some experience to fix I have decided.
Perhaps Bitron is the lubricant to use as it is ionically attracted to metal and does not attract dust. It may however spread too far as it creeps at a molecular level!

Jeremy, September 2014

The trouble with lubricants on delicate equipment is that dust gets stuck in the oil and the mechanism jams up.

I no longer have one of these Video tape machines but when I needed to do maintenance on my old one I used meths - methylated spirit - denatured alcohol - and a lint free cloth.

Take care - some people have acid perspiration - if that's you your fingerprints might leave a trace of where you've touched the metalwork.

I would avoid lubricants - perhaps you might wipe your hand over your hair and pick up some natural oil and wipe the parts with your fingers if you felt lubrication was needed - but I'd err on the side of no lubrication.

YouTube has several videos about cleaning VCR machines.

Good luck...

Robert Peston, September 2014