Why don't we say Avinu Malcanu on Tisha B' Av?
The Artscroll Siddur says "on fast days that tachanun is omitted...", is there any other reason?
The main reason that I recall is because of the pasuk in Eichah (1:15) that refers to the day as "moed"
קָרָא עָלַי מוֹעֵד לִשְׁבֹּר בַּחוּרָי
I don't remember why that forces a connection.
My rav explained that Tachanun (which, is part of your question, per se, not necessarily the direct reason why Avinu Malkeinu) is omitted on occasions of either "extreme" - extreme joy or extreme sadness.
Yom Tov is "extreme joy" so Tachanun is omitted.
In a house of mourning during the week of shiva, Tachanun is also omitted. This is a location & time of extreme sadness.
If Tachanun is omitted in a house of mourning where one person is mourning, kal vachomer it should be omitted when all of B'nei Yisra'el is mourning!
Edited to change my answer. Here is a link that explains why we don't say Avinu Malkeinu on Tisha B'Av.