Interesting question and it is true that artscroll's biography of R Moshe Feinstein doesn't address the question explicitly.
Here are a few relevant statements showing R Moshe's position. From the statements below I perceive a "positive-neutral attitude", for sure without any virulence against the State but also not proactively recommending alyah. One should remember however that R Feinstein passed away in 1986, more than 30 years ago, and hadn't seen the growth of religion (both religious Zionists and haredim) that occured since then in Israel.
R Chaim Jachter brings up his position on alyah
Rav Moshe Feinstein was asked whether one should move to Israel in
accordance with the view of the Ramban (to BeMidbar 33:53 and Mitzvah
4 of the positive Mitzvot omitted by the Rambam in his enumeration of
the 613 Mitzvot), who asserts that even in "our days," every Jew is
required to live in Israel. On the other hand, perhaps one should one
follow the opinion of Rabbeinu Chaim Cohen (cited in Tosafot Ketubot
110b s.v. Hu Omeir and Mordechai Ketubot number 313) that the Mitzvah
to live in Israel does not apply today. The latter is of the opinion
that since the journey and subsequent life in Israel is fraught with
danger and since it is difficult to fulfill the Mitzvot HaTeluyot
BaAretz (commandments associated with the land of Israel), there
exists no Mitzvah to live in Israel “today” (in the thirteen century).
Rav Feinstein argues that even though most authorities agree with the
Ramban that one fulfills a Mitzvah by living in Israel today, there is
no obligation to move to Israel. Rav Feinstein feels that the Ramban
and those who agree with him believe that if one moves to Israel he
has fulfilled a Mitzvah (Mitzvah Kiyumit) but that there exists no
absolute obligation to do so (Mitzvah Chiyuvit). Rav Moshe concludes
that since no one rules that there is an absolute obligation of
Aliyah, Rabbeinu Chaim Cohen's opinion should certainly be considered
when contemplating moving to Israel.
Rav Feinstein seeks to prove this
point from the fact that the Rambam (Hilchot Melachim 5:9) writes that
it is prohibited to leave Israel but does not state that one is
prohibited to reside outside Israel. If an obligation to move to
Israel exists, writes Rav Feinstein, then the Rambam would have
recorded a prohibition to live outside of Israel. Rav Feinstein
concludes that since there is no obligation to move to Israel even
according to the Ramban, one must certainly consider Rabbeinu Chaim
Cohen's concern that one will not fulfill the Mitzvot HaTeluyot
R Gil Student brings up his position on the prayer for the State of Israel
R. Moshe Feinstein was asked about the prayer for the state of Israel.
He said that it should be modified to indicate a Hopeful Zionist view,
instead of a Messianic Zionist approach. The text, as he recommended,
is as follows: “Our Father in heaven, the rock of Israel and its
redeemer, bless the state of Israel that it become the beginning of
the sprouting of our Redemption (she-t’hei reishis tzemihas
yeshiva.co (and revach) discuss his position on an Israeli flag inside a synagogue, found in Igrot Moshe OC 1:46
For example, in one of the few halachic responsa which touch upon the
rabbinic evaluation of modern g'dolim regarding the State, Rav Moshe
Feinstein replies to a question regarding a shul where they wished to
place an Israeli flag (together with an American flag), next to the
Ark as follows: "And even though those who made this flag and symbol
of the State of Israel were bad people (רשעים), in any case they did
not consider it (the flag, A.C.) to be a holy item, which if they had
done so, would have led to suspect that it is like idolatry... (but)
it is like every secular object... and if it was possible to dismiss
the entire matter of the flag without causing an argument, so there
will not be any memory of the actions of the bad people, this would be
the correct thing to do, but Heaven forbid causing an argument about
this". Rav Moshe infers that the Israeli flag should not be flown,
even outside the shul!
See also the interesting footnote 3
This letter was written in the year 1957/5717, and it is possible that
with the changes in the State – a general improvement in Israeli
society regarding its view of Torah and mitzvot (compared with the
days of Ben-Gurion and his quest to forcibly create a "new Jew"), the
ba'al t'shuva movement, the entrance of even the charedi parties into
all of the recent governments - Rav Feinstein may have changed his
view on the State of Israel accordingly. So it seems from his response
to his grandson who enlisted in the Israeli army, which is published
in T'chumin 5, pp.11, and so it seems also from his response in Or.
Ch. 4, 70, 11, which was written in the year 1979/5739.
This is a topic that certainly merits further study.