Ng and Schauenburg proved that the kernel of a $(2+1)$-dimensional topological quantum field theory representation of $\mathrm{SL}(2, \ints)$ is a congruence subgroup. Motivated by their result, we explore when the kernel of an irreducible representation of the braid group $B_3$ with finite image enjoys a congruence subgroup property. In particular, we show that in dimensions two and three, when the projective order of the image of the braid generator $\sigma_1$ is between 2 and 5 the kernel projects onto a congruence subgroup of $\mathrm{PSL}(2,\ints)$ and compute its level. However, for each odd integer $r$ equal to at least 5, we construct a pair of non-congruence subgroups associated with three-dimensional representations. Our techniques use classification results of low dimensional braid group representations and the Fricke-Wohlfarht theorem in number theory, as well as Tim Hsu's work on generating sets for the principal congruence subgroups of $\mathrm{PSL}(2,\ints)$.