Can one get a haircut and dye one’s hair on Ta’anit Esther?
Yes, this is permissible.
There is no prohibition on taking a haircut on a fast day.
Some write that a haircut should preferably be avoided. The source for this is the Gemara in Rosh Hashanah (18), where Rabbi Yehoshua took a haircut to show his disapproval of the enactment of a fast day on Chanukah.
This may indicate that it is somewhat improper to take a haircut on a fast day, for otherwise why should the haircut be an expression of disapproval. Based on this, the Ruach Chaim (Siman 566) writes that it is not right to take a haircut on fast days.
The Turei Even, however, points out that we do not find a restriction on haircuts for fast days other than Tisha Be’Av. With regards to Taanit Ester one could certainly take a haircut, as this is not a fast of sadness and mourning, rather of tefilla and remembering the miracles of Purim.