I wouldn't tile over lineolum.

Don't do it. I refused to hire a contractor who advised me to do just this. It's a cheesy shortcut.

I don't think it's a good idea either, especially if it's vinyl and not actual linoleum. Vinyl has a lot of give.

This is copied and pasted from the flooring page of my website:

This is a subject that's very hotly debated within the industry. It seems every manufacturer has a thinset that they say can be used over resilient flooring (sheet vinyl) or vinyl tile. The problem with that is twofold. With all the different types of vinyl flooring out now, it's tough to know which ones can, and which ones can't be tiled over. Cushioned vinyl's are out. They can compress, causing voids under the tile, ultimately causing failure. Also, not all sheet vinyl is glued solid to the floor. A lot of times, vinyl installers will only glue the edges and spot glue in the center, and if you thinset the tile to it, it won't stay very long. In my opinion, vinyl tiles are out, as well, due to the fact that the thinset bond is only as good as the bond of the surface under it, and I've seen vinyl tiles ( especially peel and stick) let go way too easily. The same is also true for sheet vinyl. The second problem with tiling over vinyl is the underlayments used for vinyl installation. Normally, in woodframe construction, there are one of several underlayments used-- luan, 1/4" particle board, 1/4" plywood, and sometimes even MDF (medium density fiberboard). ANY of those in a tile subfloor is a guaranteed failure. 1/4" plywood has a nasty habit of delaminating (the layers come apart). Luan has the same problem, plus it can compress to 1/2 its original thickness from normal residential foot traffic (I've seen this happen). Particle board (as well as MDF), just from humidity, can expand enough to pop tile loose. Any of those can cause failure, and unless you pull the vinyl, you don't know for sure whether the vinyl installer just went over the existing subfloor, or added one of the underlayments mentioned. All in all, it's an extremely risky installation, and my question is why, when spending the money it costs for a tile installation, would you want to gamble on it?

Also, another contractor who comes into the bathrooms forum from time to time (Bud Cline) has also written an article on the subject. I've linked it below:

Here is a link that might be useful: Ceramic & Stone Tile Over Vinyl Flooring

Also, here's a forum thread that was started as a discussion concerning Bud's article:

Here is a link that might be useful: Discussion: Tile Over Vinyl Flooring, by Bud Cline

Bill,
Thank you so much for all of this information. As much as we dread it, we will do it right, and remove all existing flooring, per instructions, before we start. Thank you all so very much.