The origin of fast radio bursts (FRBs), bright millisecond radio transients, is still somewhat of a mystery. Several theoretical models expect that the FRB accompanies an optical afterglow (e.g., Totani 2013; Kashiyama et al. 2013). In order to investigate the origin of FRBs, we perform $gri$-band follow-up observations of FRB~151230 (estimated $z lesssim 0.8$) with Subaru/Hyper Suprime-Cam at $8$, $11$, and $14$~days after discovery. The follow-up observation reaches a $50%$ completeness magnitude of $26.5$~mag for point sources, which is the deepest optical follow-up of FRBs to date. We find $13$ counterpart candidates with variabilities during the observation. We investigate their properties with multicolor and multi-wavelength observations and archival catalogs. Two candidates are excluded by the non-detection of FRB~151230 in the other radio feed horns that operated simultaneously to the detection, as well as the inconsistency between the photometric redshift and that derived from the dispersion measure of FRB~151230. Eight further candidates are consistent with optical variability seen in AGNs. Two more candidates are well fitted with transient templates (Type IIn supernovae), and the final candidate is poorly fitted with all of our transient templates and is located off-center of an extended source. It can only be reproduced with rapid transients with a faint peak and rapid decline and the probability of chance coincidence is $sim3.6%$. We also find that none of our candidates are consistent with Type Ia supernovae, which rules out the association of Type Ia supernovae to FRB~151230 at $zleq0.6$ and limits the dispersion measure of the host galaxy to $lesssim300$~pc~cm$^{-3}$ in a Type Ia supernova scenario.
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