Together with previous research by Ulrich Kohlenbach ([1]), a recent paper by Dag Normann and myself ([2]) provides the following pretty sharp answer.  

In a nutshell, coding continuous functions in L$_2$, the language of second-order arithmetic, does not change the RM of WKL$_0$, one of the Big Five systems.  By contrast, coding Riemann integrable functions (=continuous almost everywhere and bounded) dramatically changes the logical strength of basic theorems, like **Arzela's convergence theorem for the Riemann integral** from 1885.  

In more detail, Kohlenbach shows in [1] that in RCA$_0^\omega +$ WKL, for any third-order $Y:\mathbb{N}^\mathbb{N}\rightarrow \mathbb{N}$ that is continuous on $2^\mathbb{N}$ (via the usual epsilon-delta definition), there is an RM-code $\alpha$ such that $(\forall f \in 2^{\mathbb{N}})(Y(f)=\alpha(f)$.  This is based on a construction by Dag Normann and readily generalised to e.g. $[0,1]$.

In this way, as long as WKL is available, it does not matter whether one formulates a given theorem about continuous functions with or without codes.

Now consider Arzela's 1885 convergence theorem for the Riemann (see [0]):

Let $f$ and $(f_{n})_{n\in \mathbb{N}}$ be Riemann integrable on the unit interval and such that $\lim_{n\rightarrow \infty}f_{n}(x)=f(x)$ for all $x\in [0,1]$.  If there is $M\in \mathbb{N}$ such that $|f_{n}(x)|\leq M+1$ for all $n\in \mathbb{N}$ and $x\in [0,1]$, then $\lim_{n\rightarrow \infty}\int_{0}^{1}f_{n}(x)dx=\int_{0}^{1}f(x)dx$.


With codes, this theorem is provable in WKL or weaker.  Without codes, this theorem can be formulated in third-order arithmetic, but is not provable from third-order comprehension Z$_2^\omega$, which implies full second-order arithmetic Z$_2$. The stronger **fourth order** system Z$_2^\Omega$ does prove the above convergence theorem.


Here, Z$_2^\omega$ is Kohlenbach's RCA$_0^\omega$ extended with **third-order** functionals $S_k^2$ that decide $\Pi_k^1$-formulas (the latter formulated in $L_2$).  Such functionals $S_k^2$ are highly similar to the functionals $\nu_k$ in [0b, p. 129].  Moreover, Z$_2^\Omega$ is RCA$_0^\omega$ plus Kleene's (fourth order) $\exists^3$. Note that Z$_2^\omega$ and Z$_2^\Omega$ are conservative extensions of Z$_2$.  


In this light, the minimal comprehension axioms required to prove Arzela's convergence theorem for the Riemann integral are *massively* different depending on whether one uses codes. Moreover, we only need to go 'continuous almost everywhere/Riemann integration' for coding to break down: no need to drag in topology or other abstract stuff.  





I note that term-by-term integration, which is similar to the above convergence theorem, can already be found in the work of e.g. Dini (1870).  


References

[0] Cesaro Arzela, Sulla integrazione per serie, Atti Acc. Lincei Rend., Rome 1 (1885), 532–537

[0b] Wilfried Buchholz, Solomon Feferman, Wolfram Pohlers, and Wilfried Sieg, Iterated inductive definitions and subsystems of analysis, LNM 897, Springer, 1981.

[1] Ulrich Kohlenbach, Foundational and mathematical uses of higher types, Reflections on the foundations of mathematics, Lect. Notes Log., vol. 15, ASL, 2002, pp. 92–116.

[2] Dag Normann and Sam Sanders, On the uncountability of $\mathbb{R}$, arxiv: https://arxiv.org/abs/2007.07560