I recently went through this. No, you need not tile under it - can actually cause problems with fit etc.

We tiled behind our chimney hood to the ceiling, but marked the location of the attachment pieces first.

Tile behind the hood. If you ever need or just want to change out the hood, you won't have to try to find one to fit within the un-tiled space.

DH always says: Tile to.

Tiled behind the hood.

Lots of previous discussions about this on this forum.

To tile or not to tile ?

We typically pre-hang the hood, leaving the fasteners extended to allow for tile/stone thickness, then remove the hood, and tile behind.

Your tile man will love you for not having to make all those cuts, plus no drilling through tile/stone or searching for studs.

Also, the hood will be easily replaceable if needed in the future.

Funny, I had a kitchen and bath place tell me I shouldn't tile behind a wall mounted toilet - that they tile around it. My tile guy was happy I didn't listen to that advice!

Tile whole wall.

Assuming you will vent outside, have the vent hole cut before you tile - and tile around the hole. If not a vented hood, you still need to tile the wall behind the hood.

We just did this:

Pre-tile — hole cut out for hood:

Post tile

Finished - post-hood install.

Jpmom...your kitchen is looking great. When is the reveal?

Thank you all! Mine is venting thru the ceiling. Should I not tile the sleeve of the chimney as in the photo above?

Lisadlu16, mine vents through the ceiling and was tiled completely behind the chimney.