Does there exist a sequence of positive numbers $a_{n}$ such that $\sum a_{n}$ is convergent and $\ln(n) a_{n}$ does not converge to zero?

Sure, there is such a sequence. Find a subsequence of $1/\log n$ whose series converges. Let $a_n=1/2^n$ unless $1/\log n$ is one of the terms of the subsequence, in which case $a_n=1/\log n$.

In this example, the sequence $a_n \log n$ does not converge, as there is a subsequence of it that is constantly equal to 1, while another subsequence converges to 0. This is no accident: If $b_n>0$ for all $n$ and $b_n\log n$ converges but not to 0, then $\sum b_n$ diverges because, if $b_n\log n\to r>0$, then $b_n>r/(2\log n)$ for all $n$ large enough.