There are such representations for $n=1,2$. However, by the Wikipedia article, it seems that there are no known parametric (polynomial) representations $P,Q,R$ such that $(P(m))^3+(Q(m))^3+(R(m))^3=3$.

Is there any $n \in \mathbb N$ (other than those trivially excluded with conguence conditions) for which it is known that there do not exist three polynomials $A,B,C$ with integer coefficients such that $(A(d))^3+(B(d))^3+(C(d))^3=n$. And are there some general conjectures about non-representability of some natural numbers? Do some heuristics suggest that it is rare to have such a representation, as we have for $n=1,2$?