I will give an example from a recent paper That We See That Some Diagrammatic Proofs Are Perfectly Rigorous by Azzouni. It has been a common opinion since Pasch and Hilbert that proofs relying on diagrams, such as proofs in Euclid, lack in rigor because certain assumptions used in them are left implicit in the proof, and do not come up in the course of it. They are "wrapped into" the diagram, as it were. But diagrams can be misleading (All Triangles Are Isosceles fallacy is a typical illustration since Klein), the argument goes, so this tacit reliance constitutes a logical gap.
But diagrammatic proofs do not amount to visual reports about the looks of a diagram. There are rules in place for inferring from diagrams, just as there are rules in place for inferring from strings of symbols in formal proofs, which are held up as the standard of rigor. And those rules are also implicit, they are metamathematical. And if they are broken by oversight (e.g. if one confuses a conditional with its converse or reuses a variable in a universal instantiation) fallacies result.
Azzouni argues that the idea that diagrammatic proofs are non-rigorous is a case of the use/mention error. That some geometric assumptions are not mentioned explicitly when manipulating diagrams is taken to mean that they are not used there either, and there is a gap. The reason this error is appealing is that mathematicians generally tend to convert implicit metamathematical rules into explicit mathematical axioms, which creates the impression that something was missing before it was done. But the "missing" content actually has no effect on rigor. As an illustration, Azzouni uses the pictorial proof of the sum of the geometric series by subdividing a unit square into rectangles, see Proofs without Words:
"A charge that’s more likely to be sustained is that something (a great
deal, maybe) in the diagram is tacit. In particular, what’s tacit are the various
conventions about (two-dimensional) space that the success of the
diagram is trading on: that divisions of the square correspond to the fractions
of 1 that are so labeled in the diagram.
[...] So my suggestion is this: the respect in which the mathematical content
in a visual proof procedure is tacit amounts only to the fact that the
content in question is used and not mentioned in that proof procedure.
I’m also suggesting (and I admit this may seem odd) that the tendency
to see diagrammatic methods as not rigorous (apart from worries about
generality described in §2) ultimately boils down to a use/mention error.
The way the error proceeds is this: One senses, or is otherwise aware of,
the substantial mathematical content being used in a diagrammatic proof.
One then assumes, since this content obviously isn’t explicit (isn’t explicated
by axioms, for example), that the proof in question must involve
missing steps involving that content. But this isn’t true if that content is
playing a role enabling the proof procedure (if that content is actually
metamathematical).
When one thinks mathematical content is missing from these diagrammatic proofs,
one mixes up use and mention: the only thing missing is meta-mathematical content. But
meta-mathematical content is also missing from ordinary language proofs—which almost
everyone agrees are rigorous. Related to this point is another one. We tacitly use our visualization faculties
to appreciate the proof-theoretic elements in diagrammatic proofs.
What goes unremarked is that we use exactly the same visualization faculties
(tacitly) to appreciate the proof-theoretic elements in language proofs
as well."
