Could someone remind me why is incorrect to switch an infinite sum and an integral? Could someone jog my memory on this?

The order of operation between an $\int$ and $\sum_{n\in \mathbb{N}}$ is not always interchangable? Note that the sum is an INFINITE sum

Why is it that $\int \sum_{n \in \mathbb{N}} \neq \sum_{n \in \mathbb{N}} \int$

Is the reason because the integral itself is a sum and the order of "summing" actually matters? (I think it's Multivariable calculus related stuff now)
It is easier and more instructive to give a counterexample using sequences: For example $$\lim_{n\to\infty}\int_0^1 n^2x^n(1-x)\,dx=1$$ even though the integrand goes to zero everywhere in [0,1].
To understand what is going on here, note that the function in the integrand has a graph which is a tall, thin peak getting taller and taller and thinner and thinner as $n\to\infty$, while approaching $x=1$ from the left.

Taking differences, you can easily realize the sequence as partial sums of a series, thus providing the counterexample you seek. To be precise, consider $$\lim_{n\to\infty}\int_0^1\sum_{k=0}^{n-1} \bigl((k+1)^2x^{k+1}(1-x)-k^2x^k(1-x)\bigr)\,dx,$$ which is just a difficult way to write the limit above.
Switching an limit and an infinite sum constitutes the interchange of limit processes.  Said interchanges often yield unexpected results.