Here's an argument for why the answer is no:

First, let's formulate the question as follows. Is there a meaningful mathematical statement that can be disproved in ordinary mathematical practice but that's a theorem of ZFC under the standard definitions? We need to rule out type mismatches (is $2 \in \pi$?) and abuse of notation (whether $\mathbb{N}$ is actually a subset of $\mathbb{Z}$ or merely canonically isomorphic to a subset of $\mathbb{Z}$), but then there won't be any such statements, of course assuming ZFC is consistent.

The reason is that the ordinary mathematical arguments will have to be based on certain axioms, and the set-theoretic constructions have been designed to satisfy those axioms. For example, you want $\mathbb{N}$ to satisfy the Peano axioms, and indeed ZFC proves that it does. Assuming ZFC is consistent, you therefore can't use the Peano axioms to disprove anything ZFC proves about $\mathbb{N}$.

Similarly, when you construct $\mathbb{R}$ you prove that it's a complete ordered field (i.e., every nonempty subset that is bounded above has a least upper bound). This is enough to do elementary analysis, so our theory isn't going to contradict calculus unless ZFC is inconsistent.

In practice, when people build up mathematics within set theory, there are only two gaps in what they care about. First, there's the behavior of $\in$: ZFC assumes everything is a set, so you always have the potential for unexpected things to be elements of each other, but this is the type mismatch issue and is easily ignored. (You might worry that what if, for example, these sentences have consequences that conflict with the Peano axioms? Then ZFC would be inconsistent, since it can prove the Peano axioms.) Second, there's the abuse of notation. The von Neumann definition of $\mathbb{N}$ is beautiful and pleasant to use, but it's not literally going to be a subset of most other constructions. If you want to be super careful, you should either harmonize all your constructions or explicitly write out inclusion maps, but a little sloppiness does no harm.

The argument I've just outlined really isn't a mathematical theorem, because we haven't defined ordinary mathematical reasoning (ZFC is the closest thing we have to a definition, but that would make this whole argument vacuous). However, it's still true in a philosophical sense.

What it comes down to is that the $\mathbb{N}$ vs. $\mathbb{Z}$ abuse of notation is harmless, and that incorrectly typed statements are never used in ordinary mathematics. (If people proved everyday theorems using $2 \not\in \pi$ as an axiom, then we would have to worry about whether that axiom was compatible with our set-theoretic constructions. However, as long as ZFC is consistent, including incorrectly typed statements together with the usual axioms cannot cause problems.)

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