Let’s start with this analysis (from Trish Glazebrook’s “Why Read Heidegger On Science?”
Heidegger is not opposed to science per se insofar as he does not reject the human project of understanding nature. The most well-known basis for dismissing him as simply “anti-science” is the claim he makes repeatedly in Was Heisst Denken? that “science does not think” (WD, 4/8, et passim). But he also says often in this text that “most thought-provoking of all is that we are still not thinking” (WD, 2/4, et passim). His objection is not so much to science as to scientism, that is, the preclusion of other ways of thinking by the representational thinking of the sciences, and the marginalization, displacement, and devaluation of other methodologies and bodies of knowledge by the scientific standard of objectivity that has become epistemologically dominant in modernity.
I haven’t read Was Heisst Denken (What is Called Thinking) — a lapse I’ll need to address — but if I can interpolate a bit, I think there are two issues involved here.
First, anyone who has spent time in academia will recognize that ‘science’ as it’s practiced in the modern era is fairly thoughtless. It has fallen heavily into what Thomas Kuhn called ‘normal science’, where most scientists aren't really concerned with scientific advancement as much as they are with fleshing out practical applications, or with trivial experiments meant to pad curriculum vitae and create personal advancement within the discipline. Most of what we find in scientific journals these days isn’t meant to be insightful, challenging, or novel; they are pro forma articles discussing boilerplate experiments testing things that everyone takes for granted. These articles are meant to gain the authors position or tenure, maybe to secure some grant money, possibly to explore industrial uses that will generate income… There’s nothing wrong with that, of course, but those kinds of articles and that kind of research can be done without creative thought, and without ever reaching outside the narrow, myopic, systematic cubbyhole of that particular scientific field.
This goes back to Aristotle’s distinction between technê and epistêmê: technê is simple craftsmanship, meant to be practical and commercially viable, while epistêmê invokes real developing understanding of a subject. For people like Heidegger the narrow objectivity of modern science — narrow in the sense that it fussily focuses in on tiny controllable contexts to the exclusion of everything else in the world — is pure technê, without any real thought or understanding behind it.
Second, this collapse into technê has created a broad societal form of scientism. People have stopped investigating, questioning, or trying to understand science, and merely believe in it: a belief that science and technology will inevitably give us what we want, and so we don’t have to worry or think about it. Every year there’s a new toy, a new convenience, a new productivity tool, a new study, and so naturally people come to believe that science will also inevitably solve (say) the problems of their love lives, or political conflicts, or bigotry, or other problems of ethics, aesthetics, etc, because, you know… Science! Further, this collapse into scientism has laid the ground for the modern anti-intellectual, anti-scientific movements. Mere belief can be merely opposed; conspiracy theory and denialism are atheism for scientism, opposing the irrational belief of scientism without ever confronting (and usually actively disdaining) science itself.
For Heidegger, I think, this kind of scientism is the root of nihilism: a blind faith in science (like blind faith in God) means that people can all sink into the tiny worldviews of their immediate perceptual lives in the belief that someone or something else will take care of questions of value (moral meaning) at the same time as whatever-it-is satisfies material, teleological ends. Perhaps the next generation of iPhones will solve the ongoing crises in the middle east… Such people don’t question the relationship of science to morality — or even whether there is such a relationship — and thus lose all sense of moral meaning by deferral and default.