I was thinking the same thing as you actually. Have you tried different detergent for the sheets and clothes? Perhaps you need a mild no scent, for sensitive skin, detergent, like Tide Free and then also use the extra rinse cycle in the washer. Also do not use fabric softener or dryer sheets, they can cause irritation in sensitive skin. I'd also make sure he's just sleeping in cotton, and having cotton sheets.
Do you bathe him nightly? If you do perhaps you need to switch to a really mild soap and/or change, or don't use, moisturizers.
Really sounds like an environmental thing. I'd definitely try something new to see if it helps.
Please update if you see some changes, hope you do!