Does there exist an irreducible non-linear polynomial $P(x)\in\mathbb{Z}[x]$ such that for any prime number $q$ there exists $t\in\mathbb{N}$ such that $q|P(t)$ ?
Also (dis)proving whether there exists such polynomial that satisfies the condition for all but finitely many prime numbers is of interest.
All I have reached at the moment has been the polynomial $P(x)=(x^2-3)(x^2-5)(x^2-15)$ which is clearly not irreducible but has no integer roots and satisfies the condition of the problem and I couldn't do much more.
As you suspected, $P$ must be linear, already for all but finitely many primes.
See: https://mathoverflow.net/questions/150810/irreducible-polynomials-with-a-root-modulo-almost-all-primes