Is it an issue to discard a menorah that has been used for the mitzvah of Neros Chanuka?
Mishna B'rura to 21:1 says that once a thing used for a mitzva is no longer usable for the mitzva, it can be discarded, but should not be discarded in a degrading manner or used for a degrading purpose. He considers deliberately throwing it onto the garbage heap as an example of discarding it in a degrading manner. [I've heard recommended that such an object be put in a separate bag and then in the garbage.] [However, see comments to this answer regarding whether a m'nora is included in this category.]
[As I wrote, he only says this about something used for a mitzva and no longer usable. I conjecture that his not considering a still-usable object is simply because such things aren't generally discarded at all, but that, if bal tashchis is not an issue, the same rule will apply.]
[As Double AA notes in his answer, this does not apply to things like m'zuzos and t'filin.]
As always, for practical matters, consult your rabbi rather than relying on what you read here.
You may discard it. The gemara (Megillah 26b) states:
A menorah is certainly in the former category, especially considering you don't actually need a menorah as you could light candles directly on a table! (Although see discussion here.)