My dishwasher is in an island, across from the sink. If I connect my dishwasher directly to the sewer line instead of to a garbage disposal, then does it still need an air gap?
You do need an air-gap. In my state it is legally required (Cal Plumbing Code 414.3)
Contrary to what you might mistakenly interpret from some other comments, it does not serve the same function as your sewer/drain vent system. It is not a "vent", it is a vacuum breaker.
The purpose of an air gap in the dishwasher line is to prevent dirty water from a backed up sink from entering the dishwasher and contaminating your clean dishes. It's a type of vent.
All drain lines in the house need some type of vent, and a drain in an island is no exception. You don't need an unsightly air gap sticking out above the counter top, but the drain does need to be vented.
This image from The Family Handyman shows one type of solution for an island vent. Requirements for venting will vary from location to location, but your hired subcontractor should know the rules in your area and understand how to properly plumb this.
Its a good idea to ask the plumber how they plan on venting and you should look at the installation so you know how it all works in case something goes wrong in the future, but that's what I would recommend with any major remodel.