I am installing marble floor tile with 1/4" cement board backer. There are severl layers of vinyl flooring. There are some pieces that do not want to come up. I have even tried burning with mapp gas torch. Can I leave it even though it is the thickness of the vinyl or do I have to buy a bigger floor dog.

Help. Wife is going nuts with the length of the renovation.

What is a floor dog? You really need to get your floor flat and level. The backer board should not act like a bridge over uneven surfaces. You probably need to try harder and maybe with a better tool. You can buy or rent floor scrapers designed to remove glued tile. You could even use an old, broad chisel and just keep hammering away. Some flooring is really pain to remove but it needs to go unless you are willing to see marble cracking.
Bob

Well this might be a little late for you, but I use an old iron on a wet towell to get up tile it seems to work pretty good.