I think I can prove the following : For every continuous function $f:[a,b]\to (0,\infty)$, $\exists n_0>1$ such that for every $n\ge n_0$, there are $n+1$ distinct points in $[a,b]$ such that the interpolating polynomial $p_n(x)$ of $f$, at those points, satisfy $p_n(x) >0,\forall x \in [a,b]$.

Firstly, let $c\in[a,b]$ be such that $f(c)=\inf_{x\in[a,b]} f(x)$. Let $r:=f(c)$. Then $f(x)\ge r >0,\forall x \in [a,b]$.

Now to prove my claim, let us recall the following standard theorem (due to Marcinkiewicz I believe ?) from the theory of polynomial interpolation:

For any function $f(x)$ continuous on an interval $[a,b]$, there exists a table of nodes for which the sequence of interpolating polynomials $p_n(x)$ converges to $f(x)$ uniformly on $[a,b]$.

And then , for these sequence of polynomials, we have that $\exists n_0 \in \mathbb N$ such that $ f(x)-p_n(x)\le |f(x)-p_n(x)|\le r/2,\forall x \in [a,b],\forall n\ge n_0$, then $p_n(x)\ge f(x)-r/2\ge r-r/2=r/2>0,\forall x \in [a,b], \forall n \ge n_0$.