  We report on measurements of the differential polarizability between the
nearly degenerate, opposite parity states in atomic dysprosium at 19797.96
cm$^{-1}$. The differential scalar and tensor polarizabilities due to
additional states were measured for the $|M| = 7,...,10$ sublevels in
$^{164}$Dy and $^{162}$Dy and determined to be
$\bar{\boldsymbol\alpha}_{\scriptscriptstyle BA}^{(0)} =
180\,(45)_\text{stat}\,(8)_\text{sys}$ $\text{mHz}/(\mathrm{V}/\mathrm{cm})^2$
and $\bar{\boldsymbol\alpha}_{\scriptscriptstyle BA}^{(2)} =
-163\,(65)_\text{stat}\,(5)_\text{sys}$
$\text{mHz}/(\mathrm{V}/\mathrm{cm})^2$, respectively. The average blackbody
radiation induced Stark shift of the Zeeman spectrum was measured around 300 K
and found to be $-34(4)$ mHz/K and $+29(4)$ mHz/K for $^{164}$Dy and
$^{162}$Dy, respectively. We conclude that ac-Stark related systematics will
not limit a search for variation of the fine-structure constant, using
dysprosium, down to the level of $|\dot{\alpha}/\alpha|=2.6\times10^{-17}$
yr$^{-1}$, for two measurements of the transition frequency one year apart.
