Regarding Rav Kook specifically, I have heard that the Chazon Ish ZT”L used to censor his Seforim by taping or marking over the anti-Torah writings in them. Of course, the Chazon Ish was more able to know what is undesirable and what is not, than the average student.
If someone was the biggest Apikores and enemy of Hashem, as long as he would “work the land” of Israel, Rav Kook considered him holy. The soccer players, mechalelei shabbos b’farhesia, were to Rav Kook, “holy”. He did not mean “Tzelem Elokim” holy, but rather, because they assisted the Zionist cause they were “holy”, regardless of their status according to the Torah.
Rav Berel Soloveichik ZT”L, son of the Brisker Rav ZT”L, used to relate to his students the Chofetz Chaim’s response when he heard of Rav Kook’s position on the Cholni soccer players.
“Kook shmook!”, the Chofetz Chaim said, dismissing both the man and the position.
The story about the Chofetz Chaim – the paragon of Shemiras Halashon himself – is easily confirmed. The person who it happened with was named Rabbi Avrohom Moshe Gorelick, father of Rav Yeruchem Gorelick ZTL, who was a talmid of the Chofetz Chaim and a Rosh Yeshiva at Yeshiva University. Rav Gorelick said the story over numerous times, as did Rav Berel Soloveichik ZTL, Rosh Yeshiva of Brisk.
Rav Yosef Chaim Zonenfeld ZTL said that he was like a person who is drunk – saying irrational, nonsensical things – and in his case, he is drunk on Ahavas Yisroel. Rav Yosef Chaim Zonnenfeld’s description of him as a “Purim Rav the whole year”. Rav Kook was considered a lone, sad case of greatness gone irrational.
A more hard-line position is found in the Teshuvos Divrei Yoel by the Satmar Rav ZT”L (CM 131), where he rules outright that it is forbidden according to Halachah to follow any Halachic rulings of Rav Kook, in any area of Torah. His reasoning is based mainly on the following sources:
1) Birkei Yosef 243:3 – It is forbidden to learn Torah or listen to psakim of any Talmid Chacham that causes a Chilul Hashem.
2) Responsa Bais Shlomo YD II:101 – Any rabbi who misleads the public into sinning is forbidden to be a rabbi, and if he is a rabbi must be removed.
3) Chasam Sofer CM 163 – Any Min (i.e. Apikores), it is forbidden to hear any Torah from him.
4) The Gemora (Shabbos 116a) says that a Sefer Torah that is written by a Min must be burnt, even if the Sefer Torah is 100% proper with nothing changed in it. The reason, says the Rambam, is because we do not want and remnants of the acts of Apikorsim.
So since the Satmar Rav held that Rav Kook was an Apikores, and certainly that he mislead the masses with his Zionist teachings, he is in the category of all of the above.