There are countably many polynomials with integer coefficients of degree at least $2$; write them all in a sequence as $P_1,P_2,P_3,...\ $ For every $k\ge 1$, the set $A_k$ of all positive integers which are not the values of any of $P_1,...,P_k$ has a divergent sum of reciprocals: $\sum_{a\in A_k} 1/a=\infty$. Consequently, we can find pairwise disjoint, finite subsets $A_k'\subset A_k$ such that $\sum_{a\in A_k'} 1/a>1$. Now let $A:=\cup_{k\ge 1} A_k'$. By the construction, the series $\sum_{a\in A}1/a$ diverges, and for each $k\ge 1$, the image of $P_k$ is disjoint with $A_k'\cup A_{k+1}'\cup\ldots$; hence, has a finite intersection with $A$.
Indeed, a slight variation of this argument shows that there is a subset $A\subset{\mathbb N}$ of asymptotic density $1$ such that every non-linear polynomial with integer coefficients represents a finite number of elements of $A$ only.