This is from logic only as I have not seen a discussion other than those who discuss what would have happened had Adam managed to refrain from eating the fruit. One could say that the fact that it is not mentioned again after the chait, implies that it was no longer relevant. Indeed, the only reason given for the expulsion and the guarding of the Garden is the Aitz Hachaim. Apparently, there was no problem if Adam would have eaten more or if he had refrained in the future from eating of the Aitz Hadaat.
Note that the mitzvos that apply to both Bnai Noach and Bnai Yisrael are only those that were repeated at Sinai. Those that were given before Har Sinai and not repeated were applied only to Bnai Yisrael. (Artscroll Sanhedrin 59a3). Aitz Hadaat was never mentioned and therefore would have been one of the 613 mitzos in the same way that gid hanashe is applied only to Bnai Yisrael (Art Scroll Sanhedrin 59a3 note 23). Since it is not considered one of the 613 mitzvos, then it could not have applied to the Bnai Noach before Sinai and therefore was never applied to Bnai Yisrael by not being repeated.
Sanhedrin 56b (ArtScroll 56b1 note 9) points out that Adam was previously allowed to eat all vegetation (and fruits in Bereishis 1:29
וַיֹּ֣אמֶר אֱלֹהִ֗ים הִנֵּה֩ נָתַ֨תִּי לָכֶ֜ם אֶת־כָּל־עֵ֣שֶׂב |
זֹרֵ֣עַ זֶ֗רַע אֲשֶׁר֙ עַל־פְּנֵ֣י כָל־הָאָ֔רֶץ וְאֶת־כָּל־הָעֵ֛ץ
אֲשֶׁר־בּ֥וֹ פְרִי־עֵ֖ץ זֹרֵ֣עַ זָ֑רַע לָכֶ֥ם יִֽהְיֶ֖ה לְאָכְלָֽה:
Thus Bereishis 2:16
וַיְצַו֙ יְהֹוָ֣ה אֱלֹהִ֔ים עַל־הָֽאָדָ֖ם לֵאמֹ֑ר מִכֹּ֥ל עֵֽץ־הַגָּ֖ן
appears to be redundant. Rav Yochanan learns that this teaches what we call the Sheva Mitzvos Bnai Noach and applies this to all future generations. The Aitz Hadaat is given explicitly in the next pasuk and is shown as not part of the previous pasuk, but a one time restriction for Adam alone.
Art Scroll 56b2 note 16 explains why Eiver min hachai is shown by the אָכֹ֥ל תֹּאכֵֽל of this pasuk. It appears to me to have been repeated because the permission to eat meat was granted and Hashem wanted to avoid a misunderstanding that perhaps eiver min hachai was now allowed as well.
Thus, Aitz Hadaat besides being impossible, and since Adam had already been made mortal for violating it, was not part of the general pasuk and therefore did not apply after the expulsion.
We can also say that only the Aitz Hadaat was forbidden and any other tree of that type would have been permitted. However, we do not know if this tree was unique in the garden or if there were any more of that type. We do not even know anything about it other than it was somewhere in the Garden. The Torah does not describe it or say anything about it other than
But of the Tree of Knowledge of good and evil you shall not eat of it,
for on the day that you eat thereof, you shall surely die."
Additionally, we see from Would Adam and Chava have been permitted the tree of knowledge if they'd waited? that there are those who explain that Adam would have been permitted to eat from the Aitz Hadaas after Shabbat. As a result, we see that from the very beginning, this command was a temporary command, designed to last only until Shabbat. Therefore, even as it is impossible after the expulsion, it no longer would have applied after Shabbat in any case. Indeed, it was a one time test that Adam could have passed or failed, similar to Akeidat Yitzchak which was a one time event that Avraham passed.
The Tree of Knowledge
Thus the Midrash tells us that if only Adam and Eve had waited until
the Sabbath, they would have been permitted to eat of the Trees of
Knowledge and Life, and the purpose of creation would have been
complete. This is an astounding concept: lf humans could follow G‑d's
commands on an external basis, for no apparent reason, they would
develop a special capacity that would enable them to fulfill their
potential for higher knowledge. That capacity was the ability to
achieve penimiut (inwardness). With this, all experience would be
integrated; without it, knowledge would remain external and
fragmentary. With it, they could indeed become like-G‑d. Without it,
they would remain knowledge-seeking humans.
There is an idea that God wanted Adam to eat of the Tree of Knowledge
of Good and Evil, but only after first eating from the Tree of Life.
Adam needed to first internalize how to properly use knowledge before
eating of the fruit of that tree. He could have done so by not eating
the fruit of the Tree of Knowledge until God allowed it. Had Adam
eaten of the Tree of Life by obeying God, God would have allowed him
to eat from the Tree of Knowledge after the first Sabbath began.
Eating the fruit then would not have had negative consequences.