what mistakes did the Italian algebraic geometers actually make?

Of course, we all know great mathematicians who constantly make mistakes even now, and not because of foundations.

In any case, it's not like "long dead Italian algebraic geometers" is a category of people who were all uniformly bad. For example, Enriques was notoriously careless, while Castelnuovo was much more scrupulous (I may be wrong, but as far as know he has not made any real mistake). I remember reading of a competition for a paper on resolution of singularities of surface; Castelnuovo and Enriques were in the committee. Beppo Levi presented his famous paper on the resolution of singularities for surfaces; Enriques asked him for a couple of examples and was convinced; Castelnuovo was not. The discussion got heated. Enriques exclaimed "I am ready to cut my head if this does not work" and Castelnuovo replied "I don't think that would prove it either".

As for a result that was not simply incorrectly proved, but actually false, there is the case of the Severi bound(*) for the maximum number of singular double points of a surface in P^3. The prediction implies that there are no surfaces in P^3 of degree 6 with more than 52 nodes, but in fact there are such surfaces in P^3 with 65 nodes such as the Barth sextic (and this is optimal by Jaffe--Ruberman).

(*) Francesco Severi; "Sul massimo numero di nodi di una superficie di dato ordine dello spazio ordinario o di una forma di un iperspazio." Ann. Mat. Pura Appl. (4) 25, (1946). 1--41, MR0025179, doi:10.1007/bf02418077.

Fano's list of 3-dimensional "Fano varieties" (so named by V.A.Iskovskikh) missed an entire class, of genus 12 if I recall correctly. This list was made complete later by Iskovskikh and Mukai-Umemura.