There is a Halacha recorded by the Rema in Yoreh Deah (376:2) that one who is sitting Shiva does not have to stand up in deference to his father, his Rabbi or an elder, as usually required. The reason for this exemption is that we assume that they forgo their honor in light of the mourner's distress.
The question was asked whether this dispensation also applies to everyone on Tisha B'av, when all Jews sit on the floor in mourning. The Halacha answers that one is required to stand up in respect. The difference between sitting Shiva and Tish B'av is that the former is "Avelut D'Yachid" (Individual Mourning) and the latter is Avelut D'Rabim (Public Mourning). Since everyone is in mourning on Tish B'av, including his father, Rabbis and the elderly, there is no reason to assume that they forgo their honor.
SUMMARY
On Tisha B'av, one must stand in deference for his father, Rabbi or the elderly.
