Here is an argument that "most" points in the $SL(2, {\mathbb C})$-character variety $X(F)$ of the surface $F$ do not correspond to representations extendible to 3-manifold groups (as in the question).

Let $M$ be a compact oriented 3-manifold with $\partial M=F$. We then have the "restriction morphism" of $SL(2, {\mathbb C})$-character varieties
$$
r: X(M)\to X(F).
$$
The image of $r(X(M))$ is "formally Lagrangian" (more precisely, Lagrangian on the scheme-theoretic level) with respect to the standard complex symplectic structure on $X(F)$, see

A. Sikora, Character varieties. Trans. Amer. Math. Soc. 364 (2012), no. 10, 5173–5208.

In particular, $\dim r(X(M))\le \frac{1}{2} \dim X(F)$. Since there are only countably many 3-manifolds $M$ as above, the union
$$
U=\bigcup_{M} r(X(M)) \subset X(F)
$$
has empty interior (in the Euclidean topology). Thus, "most" points in $X(F)$ do not belong to $U$. I do not know how to detect **non-membership** in this union algorithmically. Since you are working over the complex numbers, you have to specify what computability even means. For instance, you can restrict to $\overline{{\mathbb Q}}$-points of the character variety (i.e. equivalence classes of representations to $SL(2, \overline{{\mathbb Q}})$); then at least one can use the classical notion of computability and your question is well-defined in this setting. (There is a silly algorithm which terminates for points in $U(\overline{{\mathbb Q}})$.). I do not even know if the membership problem in $U(\overline{{\mathbb Q}})$ is decidable.

PS. This entire discussion feels related to the proof of Theorem 1.3 in

N. Dunfield, W. Thurston, Finite covers of random 3-manifolds. Invent. Math. 166 (2006), no. 3, 457–521.

**Edit.** Here is one way to find explicit examples of non-extendible representations $\rho$, i.e. such that $[\rho]$ does not belong to $U$ (motivated by Ian Agol's answer in the case of genus $1$). I will use the fact that the variety $X(F)$ is ${\mathbb Q}$-rational, see for instance Theorem 2 in

A. Rapinchuk, V. Benyash-Krivetz, V. Chernousov, Representation varieties of the fundamental groups of compact orientable surfaces. Israel J. Math. 93 (1996), 29–71.

In other words, there exists a birational isomorphism defined over ${\mathbb Q}$, $f: X(F)\to {\mathbb C}^{6g-6}$.

Hence, instead of $X(F)$ we can essentially work in ${\mathbb C}^{6g-6}$ (with its standard rational structure).

Now, take a point $p=(z_1,...,z_{6g-6})$ in ${\mathbb C}^{6g-6}$ whose coordinates generate a field of transcendence degree $>3g-3$. (Such $p$ necessarily belongs to the image of $f$ and $f^{-1}(p)$ is a singleton.) One can find such tuples $p$, for instance, using the Lindemann–Weierstrass theorem. Then $[\rho]=f^{-1}(p)$ does not lie in $U$.