[Added disclaimer: What follows is the product of probably faulty memory combined with a limited understanding in the first place, so should be taken with a grain of salt.]
I believe that Brill--Noether of curves gives the kind of examples you are looking for. (My understanding, probably imperfect if not completely wrong, is that they made certain general position arguments about existence of linear systems that were just wrong, because they didn't realize that certain kinds of geometric condition were universal, and so, although they look special, are in fact general.)
You might try looking at the old papers of Harris (or maybe Eisenbud and Harris) about linear systems on curves.
Also, the introduction (by Zariski) to Zariski's collected works is interesting. He began in the Italian school, but then became instrumental in introducing algebraic tools.
Also, I think that the newest addition edition of his book on algebraic surfaces (a report on the results of the Italian school) has annotations by Mumford, which are very illuminating with regard to the differences and similarities between the Italian style and a more modern style.
P.S. Here's a way to imagine the kind of error one could make in general position arguments (although obviously any actual such error made by the Italians would be many times more subtle): Let $P_1,\ldots,P_8$ be eight points. Choose two elliptic curve $E_1$ and $E_2$ passing through the 8 points, and now try to choose them in general position (with respect to the property of containing the 8 points) so that the 9th point of intersection is in general position with regard to the $P_i$. This might seem plausibly possible if you don't think it through, but of course is in fact impossible, because the 8 given points uniquely determine the 9th one. (The possible $E_i$ lie in a pencil.) My impression is that the Italians made errors of that sort, but in much more subtle contexts.