Cement backer board layed over the flooring should be all you need.

Depends on what is under the linoleum. You need a nice, solid base to start with.

eg - I just started demo on my pantry/laundry room and found out there are 3 layers of flooring! I wouldn't consider any of them appropriate for tiling over.

You also need to consider the fact that older linoleum floors and the adhesive holding them down often contained asbestos. For that reason it is prudent to test a sample first before ripping and tearing. If it's present, proper removal can be somewhat expensive, so it often makes sense to nail/screw down a tile underlayment rather than doing a removal.

Yes, you can leave linoleum.

But, what is under everything is the most important question.
Joists?
Slab?
If joists, do you know how springy your floor is? Joist size would help to know this.

It is wrong to say
"Cement backer board layed over the flooring should be all you need"
because saying this does not deal with the floor deflection.

Everything WILL crack if the floor deflection is too much.
Cement backer board too.
Cement backer board and tile together.
Everything.

We had good success not removing the linoleum in our bathroom floor. The linoleum was, however, glued to a thick marine plywood which was laid over the subfloor & the joists were adequate. We then glued & screwed cement backerboard down & followed with the tile. Has been there about 8 years & still looks good. The only crack was under the toilet where I suspect we tightened the toilet a little too much when replacing it.