Generally, they are left on an altar or table for about 24 hours, and then are disposed of through the elements with a prayer of thanks. Libations are traditionally poured out upon the earth and food stuffs are either buried, burned, or cast into flowing water.
However, in modern times, the offerings can simply be thrown into the trash, or even flushed. Once the spirit has taken their essence, they are no longer needed.
Eating and drinking offerings should only be done with the permission of the spirit. In some traditions, for example, it is common for offerings to be shared. However, if food or drink is left out specifically for a spirit, if eaten afterwards, it tends to be stale or even tasteless as the spirit has drawn out its vital essence.