There is no such Mitzvah

There is no such Mitzvah to light 

      Chanuka candles in Shul.

1) Lighting Chanukah candles in Shul is not mentioned in Bavli, Yerushalmi or Rambam. Even the Tur & Shulchan Aruch don't say there is Chiyuv to light in Shul. They do 

mention it indirectly. (where to place it and to light it before Havdalah)

2) The Minhag of lighting in Shul started in the times of the Gaonim. 

3) This Minhag is one of the few  Minhagim where we make a Bracha  "אשר קדשנו במצותיו".

4) The Chasam Sofer writes (Shu”t  O. Ch,186) about all the large Shuls he has visited:

"Their Minhagim are based on the early Gaonim most of them lit the Menorah with wax candles". (not with oil)"

5) The Shamash from the Shul Menorah is considered a Ner Mitzvah and may not be used for personal benefit*.

*Mishna Veruro 674 sk 7 in the name of Maharshal