I believe the following sequence demonstrates the failure of flat limits to be stationary. This would be consistent with the natural interpretation of the quote, meaning: a current $T$ is called *stationary* if the varifold $\lvert T \rvert$ is.

(A quick side remark before the construction: on second thought whether $\partial T = 0$ or not seems seems unrelated to the stationarity of $\lvert T \rvert$. For example, a triple junction has boundary as a current, but is stationary as a varifold. A slightly simpler variant of the example below would see the current $S$ replaced with a triple junction.)

That being said, the currents in the constructed sequence $(T_n \mid n \in \mathbf{N})$ are one-dimensional cycles in the unit disc $D \subset \mathbf{R}^2$: $\partial T_n = 0$ for all $n$. They converge weakly as currents to another cycle, say $T_n \to T$ as $n \to \infty$. Most important: $\lvert T_n \rvert$ is stationary for all $n$, but $\lvert T \rvert$ is not.

To construct the sequence, let $\{ v_1,\dots,v_6 \} \subset \partial D$ be unit vectors with
\begin{equation}
v_1 + \cdots + v_6 = 0,
\end{equation}
but which do not match up into antipodal pairs. For example
\begin{equation}
-v_1 \not \in \{ v_1,\dots,v_6 \}.
\end{equation}

Let $S \in I_1(D)$ be the current supported in the union of the segments $\{ t v_i \mid 0 \leq t \leq 1 \}$, oriented so that
$\partial S = 0.$
The associated varifold $\lvert S \rvert$ is stationary by construction.

Let $L$ be the current supported in the segment $\{ tv_1 \mid -1 \leq t \leq 1 \}$, which we orient in the *opposite* direction. In other words
\begin{equation}
\{ t v_1 \mid 0 < t \leq 1 \} \cap \mathrm{spt} \, (S + L) = \emptyset.
\end{equation}
This too has $\partial L = 0$ and $\lvert L \rvert$ stationary.

The orientations are chosen so as to ensure that the current $T := S + L$ is not stationary; this is because $-v_1 + \cdots + v_6 = -2v_1 \neq 0$.

Next we consider a sequence of positive angles $\theta_n \to 0$. We use these angles to rotate $L$, forming a sequence of currents
\begin{equation}
R_{\theta_n \#} L \to L
\text{ as $n \to \infty$.}
\end{equation}
As long as $\theta_n$ is small enough that $\mathrm{spt} \, R_{\theta_n \#} L \cap \mathrm{spt} \, S = \{ 0 \}$, the cycles $T_n := S + R_{\theta_n \#} L$ are stationary. However
\begin{equation}
T_n = S + R_{\theta_n \#}L \to S + L = T \text{ as $n \to \infty$}
\end{equation}
in the current topology, which was pointed out above is not stationary.