It is taken as a fact that the Chazon Ish was highly critical of Rav Kook, going so far as to declare him a heretic with no share in the World to Come. Marc Shapiro, in his essay "Of Books and Bans" has a footnote (14) regarding this topic that presents textual and eye witness accounts that dispel this notion. This is not to say that the Chazon Ish did not disagree with Rav Kook on many issues, I am certain that he did. However, it appears that these disagreements took place within the realm of civility and respect that is due a Torah scholar.
I take issue with what Rapoport writes on p. 92, that when R. Kook passed away, R. Abraham Isaiah Karelitz, the Hazon Ish, declared that he would have no portion in the World to Come. The source for this is Aharon Rosenberg, Mishkenot ha-Ro’im (New York, 1997), vol. 3, pp. 1120-1121, who cites a well-known London anti-Zionist. This is hardly an unimpeachable reference. (This same source also claims that the Hazon Ish insisted that R. Ben Zion Uziel’s Mishpetei Uziel be left on the floor, since it is muktseh mei-hamat mi’us. See ibid., p. 1198; Elyakim Schlesinger’s haskamah to Aharon Rosenberg, Torat Emet [Monsey, 1992]). The truth is that while the Hazon Ish asserted that R. Kook’s philosophical works should not be read, he saw nothing objectionable about his halakhic writings and certainly did not regard as R. Kook as a heretic. See Shelomo Kohen, Pe’er ha-Dor (Jerusalem, 1969), vol. 2, p. 34. Indeed, one of the first things the Hazon Ish did when he arrived in the Land of Israel was to write R. Kook a letter, asking him to decide a halakhic problem he was confronted with. See R. Ben Zion Shapiro, ed., Iggerot ha-Reiyah (Jerusalem, 1990), pp. 448-449. Furthermore, it is known that when R. Kook came to deliver a talk in Benei Berak, the Hazon Ish remained standing throughout the former’s address. See Kohen, Pe’er ha-Dor, vol. 2, p. 32; R. Mosheh Zvi Neriyah, Bi-Sedeh ha-Reiyah (Kefar ha-Ro’eh, 1987), p. 247. Even with regard to R. Kook’s philosophical writings, the Hazon Ish sometimes expressed a more positive view, depending on whom he was speaking to. See Binyamin Efrati, "Shenei Bikurim Etsel ha-Hazon Ish ZT"L," Morashah 6 (1974): 62-63.
I’m very happy to read this, as I respect both rabbis greatly. It’s a shame that people want to besmirch tzdekim after they pass on. I guess that’s why we don’t have the 3rd Temple yet!