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 from February 2014.

For a related MO post, see here (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.

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 from February 2014.

For a related MO post, see here (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.