Let $X_i$ be iid samples and $$I_f = \frac{1}{N}\sum_{i=1}^N f(X_i)$$ be an estimator for the mean of $f(X)$ and $$I_g = \frac{1}{N}\sum_{i=1}^N g(X_i)$$ an estimator for the mean of $g(X)$.
How can someone construct confidence interval for the product of $I_f$ and $I_g$?
More general, how to construct confidence intervals for a function $F(I_f,I_g)$.
Delta method
Suppose $\{Z_n\}_{n=0}^\infty$ is a sequence of random vectors in $\mathbb{R}^K$ such that
\begin{equation} \sqrt{n} \left ( Z_n - \mu \right) \; \xrightarrow{d} \; \mathcal{N}(0,\Sigma), \end{equation}
for some $\mu\in\mathbb{R}^K$ and $\Sigma\in\mathbb{R}^{K\times K}$. Let $F:\mathbb{R}^K\rightarrow\mathbb{R}$ and $J(\mu) = \left . \nabla F(x) \right |_{x=\mu}$. Then,
\begin{equation} \sqrt{n} \big ( F(Z_n) - F(\mu) \big) \; \xrightarrow{d} \; \mathcal{N}\big( 0,J(\mu)^T \Sigma J(\mu) \big). \end{equation}
For a proof see theorem 3.1 in "A. W. van der Vaart. Asymptotic Statistics. Cambridge University Press, 1998".
Answer to your question
In order to answer your question, define $Z_n = (I_{f,n},I_{g,n})$ and $F(Z_n) = I_{f,n}I_{g,n}$ and then apply the Delta method.