Your doors are NOT structural and do not add bracing to the cabinet boxes, but the stiles do, so removing them is a "gamble". If the cabinets have a center shelf in a dado, it will probably be strong enough if the center stile is removed. If the cabinet back is securely mounted to the wall and the adjacent cabinets are secured to each other through the sides it will probably be OK. It's still a chance you would have to take. They won't fall off the wall, but the center shelf may sag a bit. These cabinets I made have no stiles across, but the shelves are in dados:
No dado'd shelving. Mdf shelving in peg holes. The boxes are attached to the wall though. While I REALLLLLLLLY want the open shelving look. I suppose it may not be the smartest thing. Especially for resale value. It's a very small kitchen with very few cabinets to begin with.
Most of the time the integrity of the cabinet depends on the span between the sides. If you remove the stile and you have more than 30" you may experience sagging of the components especially the adjustable shelves if you have them. If you have a long span you may need to re-enforce structure of the cabinet. Can you post a picture of what you are proposing?
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