No. Suppose that $T$ is a translation-invariant subspace of $k[x_1,\dots,x_n]$. Consider the graded ring $D=k[\frac{\partial}{\partial x_1},\dots,\frac{\partial}{\partial x_n}]$ of constant-coefficient differential operators. Then $T$ determines a homogeneous ideal $I(T)\subset D$, by letting the purely degree $j$ part $I_j(T)$ consist of homogeneous operators that annihilate all polynomials of degree at most $j$ in $T$.
An infinite descending chain $T_\alpha$ would give an infinite ascending chain $I(T_\alpha)$, which is impossible because $D$ is noetherian.