Since the oil used in the Menora is designated (“Muktze”) for the purpose of a Mitzva, it should not be used for mundane purposes. As such, the Tur (Orah Haim § 677) explains that it is proper to collect any leftover oil that may have remained after the eighth night and burn it, so that one does not accidentally use it for mundane purposes. This is also the ruling of the Shulhan Aruch (Orah Haim 677:4), which states that the oil is collected and burned as a small bonfire.
The Aharonim explain that, strictly speaking, this law only applies to the oil that was used during the requisite thirty-minute lighting period that was left over. For example, if one lit the Menora at Tzet HaKochavim and ten minutes later it was prematurely extinguished, this oil would need to be collected and burned after Hanukkah (unless it was used for the Menora on another night). If oil remained from after the thirty-minute period, however, it is not subject to burning since that is not considered to be technically part of the Mitzvah. Nevertheless, there is a custom to be stringent and burn any leftover oil, even if it was used after the thirty-minute period on any of the nights.
Rabbi Shmuel Kamanetzky (Kovetz Halahot, pg. 218) notes that since the purpose of burning leftover oil is to prevent using it for a mundane purpose, one may discard any leftover oil by other methods as well, such as pouring it down the drain in a respectful manner, or by wrapping the oil goblets up and then placing them in a garbage bag.
Summary: Any leftover oil from the Menora should either be burned or discarded in a respectful manner, such as wrapping it and placing in the garbage or gently pouring it down the drain.