  Ng and Schauenburg proved that the kernel of a $(2+1)$-dimensional
topological quantum field theory representation of $\mathrm{SL}(2, \mathbb{Z})$
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 $\mathrm{PSL}(2,\mathbb{Z})$ and compute its level. However, we
prove for three dimensional representations, the projective order is not enough
to decide the congruence property. For each integer of the form $2\ell \geq 6$
with $\ell$ odd, we construct a pair of non-congruence subgroups associated
with three-dimensional representations having finite image and $\sigma_1$
mapping to a matrix with projective order $2\ell$. Our technique uses
classification results of low dimensional braid group representations, and the
Fricke-Wohlfarht theorem in number theory.
