While the responses to this question have so far have generally (and correctly) focused on the robust nature of the edifice of mathematics as a whole, it might be worth pointing out by way of contrast that sometimes there are indeed doubts about foundational issues within a specific subfield (because, e.g., an "important" paper is known to have flaws), and that can be extremely deleterious to the field in question. Something along these lines is discussed in this Quanta magazine article: https://www.quantamagazine.org/the-fight-to-fix-symplectic-geometry-20170209/. (And I would say that what is described there is not even the most extreme example of what can happen.)