Insights from disproofs after counterexamples have been given Some conjectures are disproved by a single counter-example and garner little or no further interest or study, such as (to my knowledge) Euler's conjecture in number theory that at least $n$ $n^{th}$ powers are required to sum to an $n^{th}$ power, for $n>2$ (disproved by counter-example by L. J. Lander and T. R. Parkin in 1966).  But other conjectures retain interest, even after being disproved by counter-example, and lead to analytic disproofs and insights.  
Question
What are some conjectures, first known to be false through counter-example, but whose subsequent analytic disproofs shed novel insights into the problem?  More generally, what are the properties of known false conjectures that nevertheless garner significant interest and efforts of mathematicians to produce analytic disproofs?
 A: Legend has it that Pythagoras' conjecture that all numbers are rational was first known to be false via the counterexample $\sqrt2$. The study of irrational numbers has grown into a large and active subfield of Number Theory. 
A: This is not a perfect example of what you seek, but one could argue that
Francisco Santos' counterexample to the Hirsch Conjecture has reinvigorated
research on polytope diameter bounds. The Hirsch Conjecture says that in
$\mathbb{R}^d$, an $n$-facet polytope has diameter at most $n-d$. Santos
constructed an $86$-facet counterexample in $\mathbb{R}^{43}$.
In some sense, this was the impetus for the polymath_3 
project on the polynomial Hirsch Conjecture.
Progress from polymath_3 and other advances are summarized in this paper:

Santos, Francisco. "Recent progress on the combinatorial diameter of polytopes and simplicial complexes." Top 21, no. 3 (2013): 426-460.
  (Springer link to HTML paper.)

A: I would dispute the assertion that the Lander-Parkin counterexample to the $n=5$ case of Euler's conjecture has gathered no further interest or study. Elkies, in 1988, found the first counterexample to the $n=4$ case, and MathSciNet shows 21 papers that have cited Elkies' work. 
A: The first two things that come to my mind are actually the dual of what you asked for, namely conjectures where the counterexample did not come first, and the search for an explicit counterexample led to interesting developments.


*

*Brosnan and Belkale disproved a conjecture of Kontsevich about polynomially countable graphs, but their argument did not lead to a specific counterexample.  An explicit counterexample was provided by Dzmitry Doryn.

*That $\pi(x) > \mathrm{li}(x)$ for all $x$ was disproved by Littlewood, but without giving an explicit counterexample.  I believe that there is still no explicit counterexample, but the search for one has inspired some interesting work.
A: In number theory there are examples of inequalities that appear at first to go in one direction only but a counterexample was found (or not yet found) and in fact the inequality provably switches infinitely often. The granddaddy instance is $\pi(x) < {\rm Li}(x)$, for which no counterexample is known for $x > 10$ (and I think Gauss may have thought it always held), but Littlewood showed counterexamples occur infinitely often as $x\rightarrow \infty$. 
An instance with a known but large counterexample is Polya's conjecture on the sign of the partial sums $\sum_{n\leq x} \lambda(n)$, which you can find on Wikipedia. (The first counterexample is greater than 900,000,000, and the existence of counterexamples was proved before any were explicitly found.) The insight from the proofs, which is not at all evident from specific counterexamples, is that this phenomenon is quite general and is an effect of the existence of nontrivial zeros of $L$-functions. For instance, the reason the first counterexample to $\pi(x) < {\rm Li}(x)$ is beyond our present computational abilities is related to $\zeta(s)$ having an unusually large first nontrivial zero.
You can find recent papers about similar kinds of results by searching for titles like "prime number races".
A: 
What are some conjectures, first known to be false through counter-example, but whose subsequent analytic disproofs shed novel insights into the problem?

How about "every continuous function is differentiable somewhere"? Weierstrass gave an explicit counterexample (the "Weierstrass function"). Much later Mazurkiewicz disproved the assertion by showing that the set of nowhere differentiable continuous functions is comeager in the set of all continuous functions. I think it's fair to say that that shed a novel insight into the problem.
A: You might regard Russell's Paradox as a counterexample to the conjecture that there is no greatest cardinal, and its discovery led to ZFC whose development both embodied and led to immeasurable insight.
