I think in many ways the question answers itself in the first sentence. "... if we identify with the Torah's way of life."
Would anyone propose that that is optional? That according to the Torah it is OK to not identify with a Torah way of life? (We see its identification with a Torah way of life from the many statements in the Rambam about how the Chachamim desired Moshiach to be free to live a Torah life without interference, among other places).
But in terms of specifics, if doing something includes prayer, then it is an obligation to pray three times a day, and during the weekdays that prayer includes at least 4 blessings referencing and requesting various aspects of Moshiach and the messianic era. So right there is a straight up obligation, basic to every Jew. (This deduction presumes that prayer is a request that actually would affect the result of what is being requested).
Assuming you wanted something less obvious, there is this:
Footnote 128: In his Mishneh Torah, Hilchos Melachim 11:1, the Rambam writes:
In the future, the King Mashiach will arise.... Anyone who does not believe in him, or does not await his coming, denies not only [the statements of] the other prophets, but also [those of] the Torah and of Moshe, our teacher.
Implied in the Rambam's statement is that even if a person believes that ultimately Mashiach will come, but does not anxiously await his arrival, he is considered to have denied Judaism's fundamental creed
[Continuing in the main text]: The sincerity of this intent, however, must be reflected in the performance of activities to hasten Mashiach's coming for, as our Sages teach, "Action is what matters." Every man, woman, and child has an individual responsibility to work to bring about Mashiach's coming. No one else can shoulder this burden for him: his own efforts and energy are needed. Each of us must prepare for the coming of Mashiach by increasing his study of the Torah and enhancing his performance of its commandments behiddur, in a beautiful and conscientious manner.
Why is it that these are the activities which will hasten Mashiach's coming? -- Because they are intrinsically parallel to the manner in which Mashiach will relate to the Jewish people.
-- The Lubavitcher Rebbe
By the way, some people might claim that this idea that the Rambam is saying that there is an obligation to want/Anxiously await Moshiach is the Lubavitcher Rebbe's innovation, but I have seen Rav Yitzchok Zev Solovetchik quoted (here footnote 3) as saying the same thing.
The explicit extension into action (not really action, but rather infusing something we would already be obligated to do with that intention and making that a motivation for the action) is something that is perhaps more unique to Chabad Chassidus, although not just the most recent Lubavitcher Rebbe, as you find such a concept alluded to in Tanya as well as other places.