In Apostol, Mathematical Analysis, 2nd edition, Theorem 11.24 at page 332 states that
Let $f$ be a nonnegative function such that the integral $\int_{-\infty}^{+\infty} f(x) dx$ exists as an improper Riemann integral. Assume also that $f$ increases on $(-\infty, 0]$ and decreases on $[0, +\infty)$. Then we have $$\sum_{m=-\infty}^{+\infty}\frac{f(m^-)+ f(m^+)}{2}=\sum_{n=-\infty}^{+\infty}\int_{-\infty}^{+\infty} f(t) e^{-2\pi i n t} dt $$ each series being absolutely convergent.
The assumption on $f$ being increasing/decreasing seems unnecessary, and is perhaps needed because the author wants to avoid using Lebesgue integration. However, I could not find a reference that states the exact conditions on $f$ for the result to be true while removing the increasing/decreasing assumption, with $f$ being possibly discontinuous at the integers. Does anyone have a reference for such a result?
I found these papers which I post in case it could be useful to someone. It looks like $L^1$ and bounded variations is enough to get Poisson summation.
https://www.researchgate.net/publication/341579126_The_Poisson_Summation_Formula_for_Functions_of_Bounded_Variation
https://link.springer.com/article/10.14232/actasm-013-518-5