Some websites with christian apologists say that 'olam' doesn't mean 'eternal'and can also mean for a certain duration. But olam ve ad does mean eternal. Is there any place where Torah laws are seen as eternal in a unequivocal sense?
Also are there proofs in Tanakh that Torah will be followed after the resurrection?
You ask, "are there proofs in Tanakh that Torah will be followed after the resurrection?".
Based on Tehillim 88:6:
free among the dead, like the slain that lie in the grave, whom Thou rememberest no more: and they are cut off from Thy hand.
Rabbi Tanhum (Shabbos 30a) says that if a person dies, he is idle from Torah and mitzvot.
The Rambam, in his Mishneh Torah, Hilchos Yesodei HaTorah 9:1 states:
It is clear and explicit in the Torah that it is [God's] commandment, remaining forever without change, addition, or diminishment
There is an opinion in the Gemara (Niddah 61b) that after the resurrection, the mitzvot will no longer apply:
Once a person dies, he becomes free [ḥofshi] from the mitzvot.
This idea is taken from a pasuk in Devarim, where it states that we need to observe the mitzvot "this day" (Devarim 7:11)
Thou shalt therefore keep the commandments, and the statutes, and the judgments, which I command thee this day, to do them.
Rabbi Yehoshua ben Levi explained this pasuk to means (Eruvin 22a):
It means: Today is the time to do them, in this world, and tomorrow is not the time to do them, as there is no obligation or opportunity to fulfill mitzvot in the World-to-Come.
The idea that in this world we need to perform mitzvot, but in the world-to-come not, is explained in length by the Rabbeinu Bahya on Devarim 7:11:
Basically, our terrestrial world is perceived a the עולם העשיה, the world of “action,” as opposed to being the period in which one enjoys the reward for one’s actions.
The Rabbeinu Bahya cites a pasuk in Koheles, which states that "there is no action" in the world-to-come:
Whatever thy hand finds to do, do it with thy strength, for there is no work, nor device, nor knowledge, nor wisdom, in She᾽ol, whither thou goest.
So, the mitzvot we perform today, e.g. in this world, is being rewarded in the world-to-come. There is no need for performing mitzvot during the world-to-come according to the sources mentioned above.
Second, you ask "does olam really means eternal".
There are people that, based on a pasuk in Shemos (21:6) explains that olam (forever) is not really forever, since the servant, as talked about in Shemos 21:6, is "released" because of Shemittah.
The Ramban on this pasuk explains the meaning of the word "l'olam":
AND HE SHALL SERVE HIM ‘L’OLAM’ (FOREVER). Our Rabbis interpreted this to mean until the jubilee year.
This, however, is the only case where l'olam is not forever, but for a specific time. Elsewhere, when the Torah says forever, it really is forever.
