Your question is complex.
First of all, what is science? Not even methodologists have settled this question. Falseability, testeability, a method? (great read here).
Second, what do you mean by Marxism? Marxism is around 150 years old, and has evolved in the process.
Karl Popper argued that Marxism became a pseudoscience from the moment that their successors started to add ad-hoc modifications to justify why its predictions were incorrect, a view rejected by another philosopher of science, Thomas Kuhn.
If it comes to testeability, well, there are plenty of tests around of the labour theory of value (albeit not originally marxists, a key component imo) (e.g. here), tests about the tendency of the rate of profit to fall (e.g. here), or about Marx's understanding of crises (e.g. here). Under this grounds, it is a bit unfair to broadly define Marxism as unscientific. Keep in mind the two aforementioned caveats though.
Finally, about modern mainstream critiques to Marxism, there are of course many. Their existence however does not provide a foundation for its obsolescence per se, since every school of thought or ideology within a discipline faces criticism (see here a critique from a mainstream economist - Paul Romer - about mainstream Macroeconomics).
Here are two critiques from two mainstream Nobel prize winners that might be of interest:
- Robert Solow, reviewing the 1988's Palgrave Dictionary of Economics in the New York Times state:
there are always dissenting fringes within academic economics. Marxism is only the most persistent. [...] Nevertheless, there is usually a definite consensus - there is one now - and an accurate picture of the discipline would make that clear. It would have to give dissent a fair shake. It would have to treat mainstream ideas critically. But it should keep the various ''paradigms'' in proportion. I do not think ''The New Palgrave'' has managed to do that.
The most obvious, though not the most important, manifestation of imbalance is the large number of items devoted to Marxist themes [...] Marx was an important and influential thinker, and Marxism has been a doctrine with intellectual and practical influence. The fact is, however, that most serious English-speaking economists regard Marxist economics as an irrelevant dead end. ''The New Palgrave'' [...] gives a false impression of the state of play by this deadpan statement.
- George Stigler, also reviewing the 1988's Palgrave Dictionary of Economics:
Even if one is prepared to concede (as I certainly am not) that this emphasis on Marx and Sraffa is justified by their importance in modern
or earlier economics, that would not justify the frequent penetration of Marxian views into the treatment of what are surely non-Marxian subjects. [...]
A nonprofessional reader would never guess from these volumes that economists working in the Marxian-Sraffian tradition represent a
small minority of modern economists, and that their writings have virtually no impact upon the professional work of most economists in major English-language universities.
As a final comment, notice that one thing is for a theory/ideology to be unscientific and another is for it to be wrong. Furthermore, it is hard to imagine a theory/ideology that opposes the prevalent economic and political system to be mainstream.