Let $X_N$ be a random variable taking values in $\{1,\dots,N\}$, where $N=1,2,\dots$.
While I already have some options, I am interested in as weak as possible conditions for
$$ \lim_{N\to\infty}E[f(X_N)]=\lim_{N\to\infty}\sum_{n=1}^N f(n)P_N(n)=0, $$ where $P_N(n)=\Pr(X_N = n)$, and $f:\mathbb{N}\to[0,1/4]$ satisfies $f(n)=O(n^{-\alpha})$ as $n\to\infty$ for some $\alpha>0$.
Hence, I am looking for conditions on probability distributions $\{P_N\}_{N=1}^\infty$.
For a start, while $P_N(n)\to 0$ as $N\to\infty$ for every such $n=1,2,\dots$ that $f(n)\neq 0$ is an obvious necessary condition, I am still unsure whether it may also be sufficient; I couldn't come up with a counterexample to that.
How much do you know about convergence of probability measures and so on? One approach to proving your condition is sufficient is by compactifying $\mathbb N$ by adding a point at infinity. Then your $f$ extends to a continuous bounded function on $\overline{\mathbb N}$ and so forth.