Do you mean that you'd remove the plastic tiles and tile over the adhesive residue, or you'd tile over the whole mess? I DEFINITELY wouldn't tile over the plastic tile if they're popping off, but I wouldn't want to tile over the adhesive either except as a last resort. I don't think you'll be able to get a good result that way.

Have you gotten any quotes for removing the plaster from the tub/shower area and replacing it with new backerboard? It's not a very large area, and it may cost less than you think, especially if you hire a handyman rather than a professional tiler. It shouldn't take that long to do, either. Then you could finish the project yourself.

If you're looking to get by and you can't demo to the studs, I'd be inclined to take the plastic tiles off and clean up the wall as best as you can. The surface will probably be a bit ratty. It probably isn't something you'd want to tile over, nor is it a surface you could apply a membrane to with confidence.

I'd consider then screwing 1/4" tile backer board right over the plaster and lathe. With the cement board being supported by the plaster, there won't be any flex issues. Hold the bottom edge of the backer board off of the tub by a 1/4" or so.

I'd then use a topical waterproofing membrane over the backer board; Hydroban, RedGard, something along the lines of those products. You'd need to read the instructions for those materials and detail the installation if needed. Hint: The neater the job you do hanging the backer board, the less detailing you have to do prior to applying the membrane.

Then tile over the membrane. Hold the bottom edge of the bottom course of tile off the tub by about 1/8" or so. Later on you can caulk that gap.

This will thicken your wall a bit, by about 3/8" net. You might run into issues reinstalling your shower valve's trim kit. So be aware of that before you proceed. Some have flexibility and can handle the added wall thickness. Some can't.

At the edges of your tiled walls, there will be a gap between the back of the new tile and the surface of the old plaster and lathe due to the thickness of the 1/4" cement board. For a low-cost finish you can fill that with sanded grout. Or add a little detail by adding a tile trim piece for a tile border, or even add a painted wood trim piece.

Just a few ideas.

A typical shower surround will need 4 pieces of cement board, about $40.

Topical membrane, $50 for RedGard, $75 for Hydroban, 1 gallon will give you two coats in a tub surround.

Thinset, highly modified, $30 for one bag.

Grout, $20.

Tile, your choice.

If this will be DIY, no kidding go to the manufacturers' web sites for whatever backer board you chose, for whatever membrane you use, whatever thinset and grout, etc, and read the installation instructions.

They keep no secrets.

Best, Mongo

Going back to the studs is probably much easier than you'd think. Is the bathtub a modern one attached at the wall, or a freestanding one? Removing wall down to studs will also give you the opportunity to insulate your outside walls to a modern standard...