We have a tradition that Tisha B'Av will become a holiday in the time of Moshiach. Will that holiday also be nidcheh if it comes out on Shabbos?
The Gemara in Megillah 5a says that the reason we push off Tishah be-Av is because אקדומי פורענות לא מקדמי. In other words, since you cannot fast on Shabbos, Tishah be-av must be changed, and since it is a sad time, we push it off instead of moving it forward. Since this reason will not apply in the days of mashiach, there will be no reason to push it off.
Baised off of already set Halakha ( inlight of the fact that most likely when fast days become Yom Ttov's there will be a reinstated Sanhedrin who could change existing Halakha) The answer is simply no.
From what we know of about the 9th of av changing: "Thus saith the HaShem of hosts: The fast of the fourth month, and the fast of the fifth, and the fast of the seventh, and the fast of the tenth, shall be to the house of Judah joy and gladness, and cheerful seasons"-Zekharyah 8:19
Not giving any depths or details what these new joyful days actually are and still staying in the parameters of the currently established Halakhic system its safe to say that they like other Holidays would not get pushed off. As far as Halakhic sources as to why any Yom Ttov would be postponed, I know of none. This is all stating that the fast days become full Yom Ttov's and not remembrance days such as Purim or Hhanukah.
No one knows yet, but there will be no reason to postpone it since the only reason to postpone is that mourning is not allowed on Shabbat.