Do I need to put mortar between wood and backerboard on a mixed wood/tile floor?
Normally when flooring changes you would use a proper transition. for example where your tile meets your other room, you will want to leave a small gap between the wood and the tile and then use a transition to cover the gap. I would guess if you have a wood floor in the middle of a tile floor you're going to have to leave an expansion gap all the way around, which will be a good place for crumbs and debris to collect.

You might be able to use silicone to fill the expansion gap, as it would be able to flex when the wood expands/contracts. I don't think I would put grout there, because it will most likely crack when the wood moves.

Here is an example using a transition.



And without a transition.