For the heat equation imagine a rod that is heated on the left and cooled on the right. Now imagine that instead of a constant prescribed temperature on the left what we want is the heat to steadily rise and fall. This would be an example of a time-dependent boundary condition.
This would be written as:
\begin{align}
\frac{\partial{u}}{\partial{t}} = c\frac{\partial^{2}{u}}{\partial{x}^{2}}\\
u(x,0) = f(x)\\
u(0,t) = \phi(t) \hspace{4mm} u(1,t) = 0\\
\end{align}
where maybe we could model the $\phi(t)=\sin(t)$ so that the temperature rises and falls.
Similarly for the wave equation, imagine a string that is fixed at the right end but moves up and down on the left end.
Additional example:
Another example using the heat equation would be a rod with a point source in the middle, i.e.:
\begin{align}
\frac{\partial{u}}{\partial{t}} = c\frac{\partial^{2}{u}}{\partial{x}^{2}}\\
u(x,0) = \left\{
\begin{array}{ll}
1/2 & \mbox{if } x = 1/2 \\
0 & \mbox{else}
\end{array}
\right.\\
u(0,t) = 0 \hspace{4mm} u(1,t) = 0\\
\end{align}
Now this just has two zero Dirichlet conditions on the left and right ends of the rod. We could make this more interesting by adding a time dependent condition to the left end. Perhaps something like:
\begin{align}
\frac{du}{dx}(0,t) = \min(u,1)
\end{align}
This might represent some kind of chemical reaction of the left hand side where we initially have no flux when the temperature is zero there, but later as heat makes it to the wall it results in a flux.
