If I were doing it, I would usually prime and paint before installing the cabinets. My painter for this project says he'd prefer that I install everything before he paints.
Do I need to use a sealing primer on the new drywall behind where the cabinets will be installed or can I leave it unsealed?
Yes, you should prime the whole surface.
Sealing/priming helps prevent penetration of vapors and moisture from the air into the drywall. If you don't seal the entire surface, even though it's behind cabinets, penetration can happen - mind you it's not going to be significant but it can happen.
Plus the time involved to paint behind those cabinets that aren't there yet is minuscule.
This is unrelated but I don't understand your painter's objection. Painting a wall with no cabinets on it is orders of magnitude easier than painting a wall with cabinets already mounted. With cabinets in place, you've got to tape edges, you risk getting paint on your new cabinets, you've got a much harder time accessing the exposed wall (cabinets get in the way), and so forth.
I would strongly suggest the walls are primed and use a mold inhibitor behind the cabinetry, especially behind the sink cabinet and where the dishwasher will set. Raw drywall should be sealed before anything is installed against it in my opinion.
