  We present the strongest constraints to date on anisotropies of cosmic
microwave background (CMB) polarization rotation derived from 150 GHz data
taken by the BICEP2/Keck Array CMB experiments up to and including the 2014
observing season (BK14). The definition of the polarization angle in BK14 maps
has gone through self-calibration in which the overall angle is adjusted to
minimize the observed TB and EB power spectra. After this procedure, the QU
maps lose sensitivity to a uniform polarization rotation but are still
sensitive to anisotropies of polarization rotation. This analysis places
constraints on the anisotropies of polarization rotation, which could be
generated by CMB photons interacting with axionlike pseudoscalar fields or
Faraday rotation induced by primordial magnetic fields. The sensitivity of BK14
maps ($\sim 3\mu$K-arcmin) makes it possible to reconstruct anisotropies of the
polarization rotation angle and measure their angular power spectrum much more
precisely than previous attempts. Our data are found to be consistent with no
polarization rotation anisotropies, improving the upper bound on the amplitude
of the rotation angle spectrum by roughly an order of magnitude compared to the
previous best constraints. Our results lead to an order of magnitude better
constraint on the coupling constant of the Chern-Simons electromagnetic term
$g_{a\gamma}\leq 7.2\times 10^{-2}/H_I$ (95% confidence) than the constraint
derived from the B-mode spectrum, where $H_I$ is the inflationary Hubble scale.
This constraint leads to a limit on the decay constant of $10^{-6}\lesssim
f_a/M_{\rm pl}$ at mass range of $10^{-33}< m_a< 10^{-28}$ eV for $r=0.01$,
assuming $g_{a\gamma}\sim\alpha/(2\pi f_a)$ with $\alpha$ denoting the fine
structure constant. The upper bound on the amplitude of the primordial magnetic
fields is 30nG (95% confidence) from the polarization rotation anisotropies.
