How to remove floor tile?

Yes, you need to remove it. The idea behind using the backer board is that it is texture so the adhesive will have something to adhere to. Now that it’s been used, the adhesive from the tile you have now has clogged up many of the pores.

Cement backer board is used to create a smooth, strong surface for laying tile. It can be used to cover nearly any surface, including floors, back splashes and countertops. Its durability means that you don't normally have to replace it due to damage, but if you're replacing tile anyway, the backer board may not be worth saving.

Undamaged cement backer board does not need to be replaced. If tile has not been attached directly to the backer board, and it has not been dented or punctured, it can be reused without any special preparation. If it has been used as a backing for tile, or any substance that is permanently attached, it may not be worth trying to save. It is difficult to remove tile from a cement backer board, and even more difficult to avoid damaging the board in the process. Since backer board is relatively inexpensive, it's not normally worth the time and labor it would take to remove the tile to save it. You're better off replacing the backer board. If it has minor damage, such as a few dents and dings, you can use tile mortar to patch or even out these spots. Major damage, such as a hole or crumbling edges, means that the board should be replaced.

In your case it's better to replace and play it safe.