In general, the way that modern mathematical research is conducted isn't the way that many would assume is the ideal method of research. That is, mathematics is not the linear progression of foundational results leading to new results which one might expect (although the actual framework is very close). Instead, one often makes generally accepted assumptions which one expects will be proven at a later date as a technical lemma. So, in some cases, mathematicians may base large bodies of results on an essentially unproven conjecture (a so-called conditional proof).
This process is fine and is often done to 'get to the bigger results' without having to wade around in a pool of technical proofs which everyone generally accepts as folklore. It is also worth noting that such work based on false conjectures is not entirely useless and might be easily specialised to cases where the conjecture is true, or easily fixed by working around the false conjecture. The methods used in such a proof may also shed insight on the area, even if the conclusions prove false.
However, this approach has its flaws in that, until a conjecture is made theorem, there is always the possibility that it may be proven false and so essentially make any 'theorems' proven based on the conjecture redundant. I'm particularly interested in such cases that may have occurred in history, or which have the possibility of occurring in the future and having a large effect on existing literature. I would like to place the emphasis of this question on the specific effect that such falsifications of conjecture can have, as there are already many questions on unexpected counterexamples on the site.
I imagine there were several pieces of work assuming the Pólya conjecture before it was eventually proved false.
There are certainly whole areas of number theory based on the assumption that the Riemann hypothesis will eventually be proven true.
When the triangulation conjecture was proven false in dimension 4, there were many proofs based on existence of triangulations that could then only be stated for 'triangulable simplicial complexes'.
Similar to the Riemman hypothesis, there are countless papers based on the assumption that $P=NP$ (and indeed its converse).