Short answer: sheets, towels and clean blankets are usually fine. Duvets and pillows are usually refused — not because they're not needed, but because of hygiene regulations most shops can't get around.
Filled bedding items are harder to clean to a resaleable standard and fall under stricter regulations than flat textiles like sheets. Most charity shops simply don't have the facilities to guarantee a duvet or pillow meets the hygiene standard needed to sell it — so the safest, most common policy is to refuse them outright, even when they're clean and barely used.
Check with your local council — many run textile recycling schemes that accept filled bedding specifically for material recovery, even though a shop can't resell it. Some specialist textile recyclers and clothing banks also accept them. It's worth asking before assuming the only option is landfill.
Don't bin them. Animal shelters often need worn towels and sheets for bedding — call your local RSPCA branch or Dogs Trust rehoming centre before assuming they're only fit for the tip.
Always check the specific shop first — policies vary more on bedding than almost any other category. Our directory shows which local shops currently accept linens, updated from real visits rather than a generic national policy.