If I do not have any fuel burning devices, do I need a carbon monoxide detector?
Considering the low cost of these devices and how deadly CO can be, why wouldn't you want one?  That's like saying you don't want a smoke detector because you don't think anything will catch on fire.

Just because you don't have any of the items @Steve Jackson mentioned doesn't mean you can't be impacted by CO.  For example, maybe a delivery truck is parked and running in your driveway putting off all sorts of CO that is being sucked into your house.  Maybe your neighbours garage leaks near your air intakes.

One thing you can count on, if something ever did happen relating to CO, your insurance company would be the first one to find out you didn't have a detector!