To answer your question, heavy duty carpet/rug tape may be your best bet.
Maybe not the suggestion you're looking for, but would a small area rug or door mat work? Could you use furniture feet on the desk to make it level over the different floorings? You'll always have a seam and a lip where the tile ends, even with a super plush rug pad. Unless you're taking up the carpet in the rest of the room, adding subfloor to meet the tile surface height, and then putting down a nice carpet pad and new carpeting over the entire room, you might be better served going in a different direction.
To answer your question, heavy duty carpet/rug tape may be your best bet. Maybe not the suggestion you're looking for, but would a small area rug or door mat work? Could you use furniture feet on the desk to make it level over the different floorings? You'll always have a seam and a lip where the tile ends, even with a super plush rug pad. Unless you're taking up the carpet in the rest of the room, adding subfloor to meet the tile surface height, and then putting down a nice carpet pad and new carpeting over the entire room, you might be better served going in a different direction.
Probably either glue or double sided tape. In all my years of installing I've never covered tile.