In the univariate case, one can easily sample a distribution via random numbers $u\sim[0,1]$ and plugging into $F^{-1}(u)$.
I have a bivariate distribution constructed via Sklar's theorem on Copulas:
$$F(x,y)=C(F(x),F(y))$$
The distribution is not in closed form.
How do I sample from this bivariate distribution $F(x,y)$?
We would need more information on how you estimate/construct your copula $C$ to give any detailed advise.
In general, sampling from a copula can be achieved following a similar idea as the univariate case, but involving the conditional marginal distribution $F(v|u)$. To sample from the copula, one would take two independent uniform samples $\bf x$ and $\bf y$ of desired length $n$. For each $i \in \{1, \dots, n\}$, set $u_i=x_i$ and seek $v_i$ such that: $$y_i=F(v_i|u_i)=\frac{\partial C (u_i,v_i)}{\partial u_i}.$$
The pairs $(u_i,v_i)$ are now a sample from the copula. Applying the inverse of the marginals $F_X$ and $F_Y$ gives a sample of the bivariate distribution $F_{XY}(x,y)=C(F_X(x), F_Y(y))$. See as well the thread on this question.
