Is it an issue to discard a menorah that has been used for the mitzvah of Neros Chanuka?
You may discard it. The gemara (Megillah 26b) states:
תנו רבנן: תשמישי מצוה - נזרקין, תשמישי קדושה - נגנזין. ואלו הן תשמישי מצוה: סוכה, לולב, שופר, ציצית. ואלו הן תשמישי קדושה: דלוסקמי ספרים, תפילין ומזוזות, ותיק של ספר תורה, ונרתיק של תפילין ורצועותיהן
Our Rabbis taught: ‘Accessories of religious observances [when disused] are to be thrown away; accessories of holiness are to be stored away. The following are accessories of religious observances: a sukkah, a lulab, a shofar, fringes. The following are accessories of holiness: large sacks for keeping scrolls of the Scripture in, tefillin and mezuzoth, a mantle for a sefer torah, and a tefillin bag and tefillin straps’. (Translation from here)
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.)
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.