Theorems demoted back to conjectures
I just discovered that Wikipedia maintains a page entitled "List of incomplete proofs." Each of the more than $60$ entries is marked with these symbols:
"Several of the examples on the list were taken from answers to questions on the MathOverflow site."
For over $130$ years people have been steadily looking for a resolution to the following problem: what is the maximum number of limit cycles for the system of differential equations $x'=f(x,y), y'=g(x,y)$ where $f$ and $g$ are any real quadratic polynomials, in $x$ and $y$. In the 1950's, two Russian mathematicians (Petrovskii and Landis) wrote a paper claiming that the maximum is $3$. People tried to understand the proof but found holes in it, and attempted to fix it up. The famous Arnold was skeptical about the possibility. In the 1970's, two Chinese mathematicians (Chen and Wang) discovered a specific set of coefficients for the polynomials $f$ and $g$ and showed that this system has $4$ limit cycles. Of course, nobody tried to patch up the previous claim of max=$3$.
I thought this to be an amusing story, perhaps slightly in the direction of your question/request.
Here is one survey article on Hilbert's 16th Problem, where this issue is on p 5, section 4, problem 3.
Let $k$ be a field, and let $K=k(t)$ be the field of rational functions over $k$. Tate and Shafarevich [1] proved that if $k$ is a finite field, then for every $r$ there exists an elliptic curve $E/K$ with $\text{rank}~E(K)\ge r$. Their construction uses ideas from an earlier paper of Lapin [2], and indeed, the Tate-Shafarevich paper says that "analogous examples have been constructed over the field $k(t)$, when $k$ is an algebraically closed field of characteristic zero, by A. I. Lapin." However, various problems with Lapin's construction were discovered, so although the Tate-Shafarevich proof is fine for $k$ finite, the characteristic $0$ case, e.g., for $K=\mathbb C(t)$, is regarded as an open problem. (Technical note: In the case that $k$ is infinite, one requires that $E$ not be isotrivial, which means that $E$ is not isomorphic over $K$ to an elliptic curve defined over $k$.)
Tate, J., Shafarevich, I.R., The rank of elliptic curves. Dokl. Akad. Nauk SSSR 175 (1967), 770–773.
Lapin, A.I., Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 953–988.