So let's see your stuff!  

Thanks Woody. I have a very divisive bathroom, people either love it or hate it.
The bathroom suite is the original from 1965, the tiles are a slightly later addition from the mid 70's I guess.
I have some restoration to do in here this year. Does anyone know if it is possible to take tiles off a wall without damaging them?
I plan to take some tiles out from behind a fixed mirror to replace four that are missing in a very visible place.

I've taken them off successfully in one piece.
But it's hit and miss. They look like H R Johnson tiles, so decent quality.
The wider the bolster the better!
There's no way that I know of of 'dissolving' tile cement, though lots of water should soften it.

Thanks Robert.
I have been putting this job off for way too long but I'll call a good bathroom fitter in to take a look. Also need a new vanity top and sink.
I'll be so pleased if I can get four old tiles back in place as it will really bring the room back together. Fingers crossed.

Tile removal
Robert is right water would help soften the adhesive/cement, heat helps too with some adhesives, eg. from a heat gun or a powerful hairdryer. Remove the grout first with a scraper or a grinding tool and then try and get some moisture behind and wait a few days adding more moisture occasionally. Then try and pry them off. The first one will be the most difficult because of the angle.
It could be easy or difficult depending on the type of adhesive used and the wall structure. Some types of cement based tile adhesives on brick/concrete substrate ensure the tiles are very difficult to remove. One thing in your favour is that they have been there for about 40 years and the tile cement may have lost some of its strength and integrity.
If the substrate is plasterboard you could even scrape out the grout around your chosen tiles and then use a knife or other tool to cut through the plasterboard and remove the tiles as a single panel to be cleaned up later.

my little living room
I change it up at least once a year. This is how it currently looks.

My dear H.moon
Those tiles are fuc4ing fab. Where are the missing tiles located (within your bathroom)?
Enviously yours,
Aunt Mark

H.moon: those tiles are fanta...
H.moon: those tiles are fantastic.
Regarding removing them. This is actually something I have done before. Successfully. Do you have a hollow wall? If you have a solid wall the odds of success are much, much lower. Anyway, you can buy a hand tool that is sort of like a miniature masonry saw which is designed for cutting the grout out from between the tiles. Then you can cut the substrate and tiles right off the wall. At that point it is easy to get rid of the substrate.
If you have a solid wall you still need to take the grout out around the tiles, so it doesn't cause the face to chip when you pry them off, which might not be easy depending on how well they were stuck.

I was thinking like leif--
though if you're in the UK then you might have interior masonry walls. Actually I dunno about the UK--but the house we owned in Denmark had all brick walls inside.
(It also had tiled bathrooms very much like yours, H, but with very ugly tile the color and visual texture of algae.)

If the tiles are on...
If the tiles are on plaster/lathe or a gypsum based wall board, I agree with Leif's advice to just remove the wall substrate along with the tile itself. To remove grout I use an oscillating cutting tool like a Fein Multimaster with its grout removal blade, which will also cut through the plaster or drywall.
If it is a solid masonry wall, that is another story, as Leif said. From your image, the mirror appears to be on an interior wall at least.
edit: although if your house is like the one Spanky lived in, it might not matter if it is an interior wall, if all the interior walls are brick.

I am sad to admit that these...
I am sad to admit that these images probably mask the fact that this home is that of a hoarder. I won't be posting photos of the garage or spare bedrooms, etc.

Thank you for all the good advise.
Thank you Mark, I did actually think of you when I saw this bathroom as it reminded me of a pair of your pants.
Heres the issue. Four white tiles in the top corner, very very upsetting.
Love your style tchp.

BTW, the good tiles I want to remove are on a solid brick wall.

and I
guess you've been keeping an eye on eBay for the same tiles to turn up?

Get someone to paint the white tiles.
They're up in the corner so no one will be getting close enough to them to tell that they're not original (unless they're snooping, then shame on them! haha). The pattern is well defined and should be very easy to reproduce.
Tchp, I love your place. And I admire your ability to confine the hoarding to specific rooms. No one would ever suspect!

or
photograph them, print it out, paste it up and glaze over it?