I would go all the way up to the ceiling and have you considered keeping the edge even with the cabinets vs going to the doorway? if you want to keep it to the doorway then I would suggest going to the floor as well. what is on the other side of the doorway? if cabinets continue I would tile over and that side as well. any more pictures/views of the room?

I would not tile that little sliver. In theory, a backsplash is there to prevent soiling of the wall by errant food material. Tiling that little slice of wall serves no purpose. Yes, I know, current trends lean toward tiling almost any available surface.

By the time you place your schluter edge there, and leave the space for grout and caulk...you are looking at a very tiny strip of tile? If the painted drywall bothers you, I would try to color match some paint to your cabinet and fill in the space where the tiny tiles would go. If it still bugs...then give the tiling a go. I don't think that it needs it.

Lovely kitchen.

I agree with lucky. If the space was wider, I would say yes to tiling all the way down.

We have seen discrepancies much worse.

I think you can leave it as is.

What is your counter?

I think it would look nice tiled down to the floor

You went past the counter edge with the tiles already, I would tile to the bottom.

@Eld6161 counter is quartz.

Skipjack you nailed the issue. I probably should have started the tile at the counter edge, instead of flush with the door

Glo European.. thanks for the virtual tile job!

Tile to the floor

Glo E's photo made it easy. Tile it in.

I agree with @lucky998877 - this is not a worthwhile effort for that small sliver of space. And it could end up looking bad because of trying to jam in slivers of tile, rather than the clean look it has now. Plus, it is "backsplash" tile, which it is named for a reason. So, what is splashing back on that side of the cabinet? No, leave it alone.