While Lin Biao's example is perhaps somewhat relevant, that of Peng Dehuai (whom Lin replaced as defense minister in the late 1950s) is perhaps even more so.
Peng perhaps did not intend for his (policy) criticism to become (as) public, but Mao made it so when he shared Peng's letter. As Wikipedia recounts
In Spring 1959, PRC Defense Minister Peng Dehuai led a Chinese military delegation on a visit to Eastern Europe and the Soviet Union. Peng expressed his displeasure with the Great Leap Forward to various communist leaders, including Nikita Khruschev. In his view, the socioeconomic policies of the period undermined the economic development necessary to modernize the army. On his return to China in mid-June, Peng criticized the Great Leap Forward.
Peng's criticism culminated in his "Letter of Opinion." On July 14, Peng wrote a private letter to Mao criticizing some elements of the Great Leap Forward. In the letter, he cautiously framed his words and did not deny the "great achievement" of Mao, but meanwhile showed his disapproval for elements like the "winds of exaggeration" (i.e., over-reporting of grain production), the communal dining, and also the establishment of commune militia, which he felt would undermine the strength of the People's Liberation Army. He expressed his "confusion" towards "rather large losses" and the "epidemic of bragging" in the Great Leap Forward.Peng attributed the problems to "petty bourgeois fanaticism."
On July 23, Mao showed Peng's letter to his comrades and asked them to express their views on the issue. Peng made no further substantive argument other than for the party to immediately withdraw from political initiatives in rural areas. Peng's position found no support among other conference attendees, as it amounted to "political suicide" for the party. For example, Zhou Enlai, normally a mediator between the left and right sides of the party, was extremely critical of Peng. Additionally, Peng's position would mean de facto realignment with Soviet approaches at a time when Mao had been trying to find an independent path in terms of both foreign and domestic policy approaches.
[...] The Lushan Plenum adopted a resolution denouncing "the anti-Party clique headed by Peng Duhai."
Later on, Peng was publicly humiliated by Red Guards etc.
Granted, the disagreement that led to his downfall was not as much about military objectives as about economic policy that Peng waded into.
Interestingly enough, Peng did have some disagreements with Mao during the fighting in the 1930s, when Peng was a commanding general. Peng favored a more traditional set-piece confrontation with the nationalists, but after several defeats like that embraced Mao's guerilla approach. Although those disagreements didn't lead to personal repercussions for him, it seems. Mao even seems to have appreciated Peng's fighting spirit back then, as illustrated in a poem Mao published in 1947 (when Peng was still in Mao's favors).
N.B. Zhu De's final years were also affected by this event. He only advanced mild criticism of Peng, found so unsatisfactory by Mao that Zhu was dismissed from the Politburo Standing Committee.
As for Lin, besides the stuff about Korea which was 20-years water under the bridge by the time of his downfall, he did have more reservations about Mao's ideas, but apparently didn't publicize them.
Lin's system of indoctrination made it clear the Party was in command of China's armed forces, and Lin ensured that the army's political commissars enjoyed great power and status in order to see that his directives were followed. Lin implemented these reforms in order to please Mao, but privately was concerned that they would weaken the PLA (which they did). Mao strongly approved of these reforms, and conscientiously promoted Lin to a series of high positions.
I don't have quick access to the book that Wikipedia cites there, so I don't know how comes we nowadays know what Lin's private concerns were.