Blücher was born in Berlin. He was a member of the Communist Party of Germany until 1928, but soon rejected Stalinism and left the party in protest of its Stalinist policies. He then became a member of a small anti-Stalinist group called the Communist Party Opposition.
As a Communist (albeit anti-Stalinist), Blücher, then a university lecturer (Dozent), had to flee Germany following the rise of National Socialism. He married Hannah Arendt in France, and they emigrated to New York City in 1941.
Heinrich Blücher began teaching philosophy at Bard College in 1952, continuing for seventeen years, as well as at the New School for Social Research. Blücher died in New York. His wife Hannah Arendt was later buried alongside him at Bard College Cemetery.
Blücher encouraged his wife to become involved with Marxism and political theory, though ultimately her use of Karl Marx was in no way orthodox, as shown in such works as The Origins of Totalitarianism (1951) and The Human Condition (1958). Blücher also coined the term "the anti-political principle" to describe totalitarianism's destruction of a space of resistance — a term taken up both by Arendt and Karl Jaspers.