From the USGA, a recommendation to clean up after yourself to save the mowers.

Golf tees are usually made of wood or plastic, but regardless of the material, they are foreign objects that do not belong in the path of a carefully sharpened and adjusted tee mower. When cutting units encounter broken or discarded tees, chewing through the hard material will dull the blades and reduce mowing quality. Instead of a crisp, clean cut, the grass can be torn or ripped, which weakens plant health and diminishes surface quality.

If the course you’re playing has containers for broken tees, please try to use them. If there is not a container readily available, you can look for a garbage can or drop the fragments into your pocket or golf cart until you find a good place to dispose of them. If nothing else, it’s better to throw broken tees into the rough instead of leaving them on the tee surface because rough mowers will not be as affected by the debris. Cleaning up our tee debris helps protect equipment, makes course setup more efficient for the maintenance staff, and preserves the quality of the turf. It is a simple courtesy that benefits everyone who enjoys the course. 

The Downsides of Discarded Tees