What quotes or examples from the poem "Howl" by Allen Ginsberg help demonstrate that the author is using the poem to protest against technocracy?
Technocracy was a political movement that became briefly popular in the US during the 1930s. The basic premise behind the movement was the idea that the current money-driven government was inadequate to effectively act for and take care of its people. Instead, those who supported the technocracy movement argued that it was the scientists and engineers who should be managing the government, rather than the politicians and businesspersons.
Though the technocracy movement was popular more than two decades before "Howl" was written and it is unlikely that Ginsberg wrote the poem to specifically counter technocracy, there are a few places in Allen Ginsberg's work that make it clear he is for the freedom to rebel, and thus by extension protests technocracy.
One such portion of the poem that supports this are the lines
who bit detectives in the neck and shrieked with delight in policecars for committing no crime but their own wild cooking pederasty and intoxication,
who howled on their knees in the subway and were dragged off the roof waving genitals and manuscripts...
The "who" of the poem are the "best minds of my generation" Ginsberg refers to in the poem's first line. As "Howl" is a celebration of the counterculture movement of the 1950s (also known as the Beatnik movement) Ginsberg's mention that these "best minds" engaged in disorderly conduct is not a condemnation of rebellious actions. Rather, the poet extols the often socially inappropriate and revolutionary actions he lists in "Howl." These acts of rebellion stand in direct contrast to the values of technocracy, which as a movement "was committed to abstaining from all revolutionary and political activities" (Wikipedia).
Since technocracy advocates a socially engineered form of government that would eliminate the need for political opinions, "Howl" protests the movement by its entire classification as a revolutionary, provocative, and political poem. Almost any quote from Part I of the poem could serve to back that statement.