Q. Can you have a wedding on the night of the Tenth of Teveth, since the fasting starts only in the morning?
A. Remah (O.H. 550: 3) writes that if a chupa is celebrated on Erev Shabbos, the minhag is to daven mincha and read Vayichal first and then the chupa begins. Mishna Berura (ibid. 12) rules that the chosson should not drink from the cup of wine since it is still a fast day, it is then given to a child. On note 10 Mishna Berura explains that this taanis must be Assara Beteves, since it is the only one that can occur on Erev Shabbos.
Similarly, Tosafos (Eiruvin 40b) mentions the case of a chupa done on Assarah Beteves and the wine cup was given to a child.
Maharsham (Daas Torah 551: 2) maintains that one should avoid dancing on a chupa celebrated on Assarah Beteves.
However, Tzitz Eliezer (7: 10: 8) quotes different opinions of Poskim in regards to a chupa on that day.
Beis Avi (6 quoted in Shaar Halocho p.42), writes that the minhag is not to make a chupa even on the night prior to the fasting, unless in great need.
Ratz Katzvi (16) maintains that on this Taanis a baal nefesh should begin fasting from the night before. Weddings should therefore be avoided on that night.
Eben Pina (O.H. 45) permits performing a wedding on the night prior the Tenth of Teves only after the fact, when the date was already chosen. Lechatchila, one should avoid that date.
Horav Shlomo Miller’s Shlit’a opinion is also to avoid observing a chupa on the night of Assara Beteves, unless there is great need.

Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit’a