As mentioned above, many cotton varieties are eco-friendly. Cotton is a renewable resource to begin with and organic cotton wasn't treated with pesticides. Many brands also focus on harvesting that cotton in sustainable ways. Similarly, you might also find organic hemp or organic linen.
Another option is to find other sustainable materials. You can even find sheets made out of bamboo, as an example. Bamboo sheets can be made to be soft, breathable, moisture-wicking, antibacterial, hypoallergenic and durable. It's even known for getting softer the more times it's washed. Bamboo itself is sustainable because it grows back very quickly.
One drawback is that bamboo sheets & pillowcases can be moreexpensive. You also have to shop carefully to make sure the manufacturing process the company uses is environmentally friendly. Sometimes a lot of harsh processing goes into making sheets.
Another common term you'll come across when looking for organic sheets is Tencel - a material made from the pulp of eucalyptus trees. It's a type of sheet that can take fewer chemicals and other resources to produce. It's durable, soft and antimicrobial. However, Tencel is not as breathable as other options like cotton.
As a general note, not all eco-friendly labels are the same. Make sure the brand is well-recognized as being legitimately organic by doing some online research before you purchase. You also may want to make sure the brand carries some sort of independent verification, like if it's Fair Trade Certified or Rainforest Alliance Certified.