Recyclable, but not at home
Polyester strings are plastic, so in theory they belong in the recycling stream. In practice they are too thin and too tangled for curbside sorting machines to catch, so they fall through and go to landfill. They need a facility that handles them on purpose. That is the gap String Savvy fills.
Not recyclable, but not plastic either
Gut strings are made from animal fibre rather than plastic. They cannot go in recycling, but they also are not adding plastic to landfill. Most players use polyester or a polyester hybrid, so check your stringer's label if you are not sure what came off your frame.
Not curbside recyclable
A tennis ball is rubber bonded to felt, and that mix is difficult to separate, so balls are not accepted in household recycling. Specialist programmes do exist, and dead balls also get a second life at schools, dog rescues, and as chair leg covers in classrooms.
Usually recyclable
The clear plastic tube is generally accepted in curbside recycling once it is empty. Rinse it, take the lid off, and check your local guidance if the can has a metal base or lid, since those sometimes need separating.
Not recyclable
Used overgrips are a mix of materials, soaked in sweat, and too small to sort. They belong in the bin. The savvy move here is using fewer of them rather than trying to recycle them.