You may safely dispose of litmus paper the same manner in which you dispose of any other paper. Litmus paper is a type of filter paper coated with a "litmus" pigment solution made from organic coloring matter obtained from Roccella tinctoria and other species of lichens. Once dried and cut into strips, the paper is ready for use.
The lichen species used to manufacture litmus paper are a source of azolitmin, which is the reacting agent in the solution. The solution contains a weak diprotic acid that releases hydrogen ions in alkaline solutions and turns the paper blue. In acidic solutions, the paper turns red.
Litmus paper does not pose any significant health risks under normal circumstances. Most of the other types of dyes used to manufacture litmus are plant- or lichen-based, and some are approved for use in food colorings.
Litmus paper reacts by turning red when exposed to acids and blue when exposed to alkaline solutions. Most other types of dyes used for litmus paper are derived from lichen and are approved for use as food colorings. In addition, litmus paper is generally effective within a narrow pH range of approximately 4.5 to 8.3, so even after it has been used for testing, the pH range and amount of solution retained by the paper are generally not sufficient to pose risks from handling and do not warrant special methods of disposal.