First, three possible areas of damage (though there are surely more): 

1. Subsequent results that make use of these "proofs" (especially when the claim is not true);

2. Using the *methods* of the incorrect proofs, when, in fact, this is where the problem lies;  and 

3. Causing others to lose trust in the institution of mathematics (e.g., questioning rigor more broadly). 

Second, an explicit example: Du-Hwang's proof of the Gilbert-Pollak Conjecture, which was later shown to contain a serious gap. The go-to for a "proof" of it was a text by Ivanov and Tužilin, but since the error in the proof has been discovered, those two have gone on to explain not only where the Du-Hwang proof went wrong, but also why attempts to patch it up have been unsuccessful. To this latter end, see their arXiv note **[here][1]** from February 2014.

For a related MO post, see [**here**][2] (where I believe the top comment is from Ivanov) and a link to the note mentioned above (which contains references for further reading).

More generally, one might reasonably expect that realizing a proof is wrong took some insight, and where there is insight, it seems quite possible that there will be an *inspiration for new work*. Whether or not that work will lead to newfound success is sure to occur on a case-by-case basis; I'm not sure that the error in the Du-Hwang proof has led to anything of great import at this time, though it has renewed a bit of *interest* in the area of Steiner minimal trees.


  [1]: http://arxiv.org/abs/1402.6079v1
  [2]: https://mathoverflow.net/q/144081/22971