It will not affect the structure. I took out a lot of the cabinetry in my trailer -- the cabinets were secured to the frame with staples and glue! Not part of the framing, in other words. (Don't try this in your stick and brick house -- sometimes, things that look extraneous are actually "bearing walls.")
Just be gentle in removing the old cabinets -- if you are too vigorous, you can peel off the wooden panels that are glued to the foam in the walls. Don't ask how I know this.
If you want to cut the cabinet frames flush with the wall, use an oscillating tool with a metal/wood blade. They are very inexpensive and very useful tools. You need a hybrid blade because you will encounter both wood and the metal staples.
2012 Fun Finder X-139 "Boondock Style" (axle-flipped and extra insulation)
2013 Toyota Tacoma Off-Road (semi-beefy tires and components)