Surely this is not an insurmountable problem.
Fridges are made to be moved.
Simply hire some muscles and have it moved to another area. Open the fridge door wide and slide the glass shelf right out.
Easy, peasey, Japanesy. 
I wouldn’t bother, go out and buy a decent sized fridge, how can you get 24 stubbies of beer in that tiny thing?
You can then put it out on the verge for some other needy sod to struggle with.
I know – you ‘shelved’ the original idea of even trying to move the offending shelf, then went online to ao.com and ordered a new fridge/freezer!
Meanwhile, waiting for ‘next day delivery’ you thought you would give members some fun and have managed to get some 60+ posts in reply on your thread, as you usually do! 
It would have cost me more than the fridge- freezer is worth to do that!!
Oooh Bruce, l must get down to the supermarket at 10am today, not to buy food but load up with the essentials… like beer, stubbies, tinnies or whatever they’re called!
Anyway, you still haven’t guessed how l did it!!
Stop hallucinating and tell me how l did it!
I can’t believe none of the strapping men… or dainty women on here, haven’t guessed!!
It was so simple, now put your thinking hat on!
Sixty-six posts, now sixty-seven, it’s going well!
OK a second chance to guess. If that was me then I might have rolled up a towel or similar, placed it beneath the uppermost end of the shelf, in case it broke, and given it a strong, upward whack with my hand or fist to dislodge that end of the shelf.
Any good Artangel?
Pleased you managed to remove the shelf but more interested in your plans for my favourite blue cheese, Art?
You were requested to tell us the answer in post 60 Art, times up I think so put us out of our misery please…
Sort of but there’s more.
LT, St Agur blue cheese is lovely isn’t it? I like it on sea salt crackers, even better with a glass of nice red wine!
See you shortly!
Mags, Yes, l was requested to tell… but by only one member.
I presumed no one else was interested so l left it!
You’ve got a long journey!
Talk about intrigued to know the answer to this conundrum.
But there’s more? 
Seventy-five posts now, it might just make the 100 any time soon!
Ok this is what l did, it’s so simple really.
I turned off the fridge as TedC advised.
I then used vegetable oil to grease the side of the glass shelf and the protrusion where l wanted to get the shelf over.
With an old fashioned flat bladed dinner knife, l eased the blade between the glass and the ridge of the plastic protrusion to make a straight side.
I then firmly and systematically bumped the glass up over the blade. I had to use all my strength as it kept slipping back down. I was also scared that l might break the glass and lose my hand!
Anyway, after a few hard bumps, it moved into a position where l could get it out and put it where l should have put it in the first place.
I nearly had that correct then! Personally I always use washing-up liquid for anything like that, it washes off easily. A rolled-up towel or similar would perhaps have alleviated some of the worries about damaging your hand.
Now it’s sorted and also gave members the chance to exercise their brains, me included in that! Only one final comment to make – good you got it ‘sorted’, don’t do it again!
Brute force will out then Artangel - well done.
Sounds like brute force that got it jammed in the first place :shock: 
Remains a bit of a mystery to me how you managed to get it in in the first place. Removal should surely have been the reversal of insertion.