This is an attempt to go further than this answer.
Essentially, we have either a summation of an integral:
$$\sum_x{ \left( \int{ f(x)dx } \right) } \tag{1}$$
...or an integral of a summation:
$$\int{ \left( \sum_x{ f(x) } \right) dx } \tag{2}$$
The answer, without going into detail, states that we can generally interchange the summations and integrations if $f(x)$ is positive. I'm wondering if it's ever possible to interchange summation and integration if $f(x)$ happens to be negative for some values, and what these cases are. I've shown each case separately as (1) and (2), in case that one direction is different than the other.
Yes, just look at the answer just above the one you gave the link to : if $\int \sum |f_n| < \infty$ or $\sum \int |f_n| < \infty$ (which are two equivalent conditions according to Tonelli's theorem as stated in the answer), then you can interchange summation and integral.
In usual settings (domain of $f$ is in a complete measured space, $f$ is valuated in a euclidian space, etc...), you can derive it from either using Dominated Convergence Theorem, or Fubini/Tonelli theorems where one of the measured space is $\mathbb{N}$ with the counting measure. AFAIK, there is no other well-known and general theorem with such simple hypothesis that ensures interchangeability.