I guess this is standard and there is a citable reference. I think the following is an argument which only uses the formalism (e.g. also works in the étale case).
Firstly, if $X \to Y \to Z \stackrel{+1}{\to}$ is a distinguished triangle with $X, Y$ of wt $\ge 0$, then $Z$ is of weight $\ge 0$. (This is easy if one thinks about Frobenius eigenvalues.)
Dually, if $X \to Y \to Z \stackrel{+1}{\to}$ is a dt with $Y, Z$ of wt $\le 0$ then $X$ is of wt $\le 0$.
Now the constant sheaf $k_Y$ on $Y$ is of wt $\le 0$. Hence the dualizing sheaf $\omega_Y$ on $Y$ is of wt $\ge 0$.
Let $f : X \to Y$ be as in your question. Consider the distinguished triangle $K \to f_!f^!\omega_Y \to \omega_Y \stackrel{+1}{\to}$ (where $K$ is defined as the shift of the cone over the adjunction morphism $f_!f^! \to id$). The above remarks imply that $K$ is of weight $\ge 0$ (as $f^!\omega_Y = \omega_X$ and $f_! = f_*$ preserves wts $\ge 0$).
Pushing to a point we get a long exact sequence
$\dots \to H^k(Y,K) \to H^k(X,\omega_X) \to H^k(X,\omega_Y) \to H^{k+1}(Y,K) \to \dots$
now everything in $H^{k+1}(Y,K)$ is of weight $\ge k+1$ (because $*$-pushforward preserves wt $\ge 0$). We conclude that we have a surjection
$gr_W^kH^k(X,\omega_X) \to gr_W^kH^k(X,\omega_Y)$
Finally, because $H_k^{BM}(X) = H^{-k}(X,\omega_X)$ (Borel-Moore homology) the result follows.