The Fisher information theorem and the law of large numbers tell us that $\tfrac{1}{n}\oI_n(\bts) \inP \fI(\bts)$. In practice, however, we never know $\bts$ and must evaluate the information at an estimate instead. The following theorem shows that this makes no difference asymptotically, provided that the estimate is consistent.
Theorem: Suppose regularity conditions (A)-(C) are met. Then for any consistent estimator $\btt$,
\[\tfrac{1}{n}\oI_n(\btt) \inP \fI(\bts).\]Proof: Because $\bTs$ is open and $\btt$ is consistent, $\btt$ lies in a neighborhood $N_r(\bts) \subset \bTs$ with probability tending to 1, which is all we need to take Taylor expansions about $\bts$.
Consider element $jk$ of the information contributed by observation $i$, $\oI_i(\bt)_{jk} = -\partial^2 \ell_i(\bt)/\partial\theta_j\partial\theta_k$. This is a scalar function of $\bt$ whose gradient consists of third partial derivatives of $\ell_i$, each of which is bounded by $M(x_i)$ on $\bTs$ by C(iii); thus the gradient has norm at most $\sqrt{d}M(x_i)$. By a Taylor expansion, there is a $\bar{\bt}$ on the line segment between $\btt$ and $\bts$ (and therefore in $N_r(\bts)$) such that
\[\begin{alignat*}{2} \abs{\oI_i(\btt)_{jk} - \oI_i(\bts)_{jk}} &= \abs{\nabla \oI_i(\bar{\bt})_{jk} \tp (\btt-\bts)} &\hspace{4em}& \href{taylor-expansion.html}{\text{Taylor}} \\ &\le \sqrt{d} M(x_i) \norm{\btt-\bts} && \href{cauchy-schwarz.html}{\text{Cauchy-Schwarz}} \end{alignat*}\]Summing these contributions and dividing by $n$,
\[\begin{alignat*}{2} \norm{\tfrac{1}{n}\oI_n(\btt) - \tfrac{1}{n}\oI_n(\bts)}_{\max} &\le \tfrac{1}{n}\sum_{i=1}^n \norm{\oI_i(\btt) - \oI_i(\bts)}_{\max} &\hspace{4em}& \text{Triangle inequality} \\ &\le \sqrt{d} \left\{\tfrac{1}{n}\sum_{i=1}^n M(x_i)\right\} \norm{\btt-\bts} \\ &= O_p(1) o_p(1) && \href{weak-law-of-large-numbers.html}{\text{WLLN}}, \Ex M(X) < \infty; \href{consistent.html}{\btt \text{ consistent}} \\ &= o_p(1) && \href{o-notation.html}{O \text{ notation}} \end{alignat*}\]Finally, $\tfrac{1}{n}\oI_n(\bts) \inP \Ex\oI(\bts \mid X) = \fI(\bts)$ by the WLLN and the Fisher information theorem.
Note that the bound above does not depend on where $\btt$ falls within $N_r(\bts)$: the single integrable bound $M(x)$ makes $\tfrac{1}{n}\oI_n$ uniformly continuous over $\bTs$ (with probability tending to 1), and this is what allows the information and the point at which it is evaluated to change simultaneously. The continuous mapping theorem alone would not be enough here, since $\tfrac{1}{n}\oI_n$ is a different random function for each $n$.
Fisher information
The analogous result for the Fisher information, $\fI(\btt) \inP \fI(\bts)$, follows from the continuous mapping theorem provided that $\fI(\bt)$ is continuous at $\bts$. Interestingly, this is not guaranteed by conditions (A)-(C). The difficulty is that
\[\fI(\bt) = \int \oI(\bt \mid x) \, dP(x \mid \bt),\]so as $\bt$ changes, both the integrand and the distribution we are integrating over change. Condition C(iii) controls the integrand, but only through a bound $M(x)$ that is integrable with respect to $P(x \mid \bts)$; it says nothing about how $P(x \mid \bt)$ changes with $\bt$, so we have no dominating function with which to pass the limit under the integral sign.
In practice, the continuity of $\fI(\bt)$ is rarely in doubt and is usually checked directly; it is typically available in closed form. For example, in an exponential family with natural parameter $\bt$, $\fI(\bt) = \nabla^2\psi(\bt)$, and the cumulant generating function $\psi$ is infinitely differentiable on the interior of the natural parameter space.