Can discrete time series be non-stationary? We all know about time series that grow over time, but it seems like we only ever see continuous values such as the plot shown below. 
But is there such thing as a time series that might start out with balanced mix of class 0 and class 1, but eventually turn into mostly class 1's?
 A: Per the title, certainly discrete series can be nonstationary.
The total count of events in a Poisson process is one example - let $N_t$ be the number of events by time $t$ and consider the time series formed by looking at $N_t$ at integer times: $N_1$, $N_2$, $N_3$,...
We have that $E(N_t)=\lambda t$, $\text{Var}(N_t)=\lambda t$; so the above time series is not even weakly stationary.

But is there such thing as a time series that might start out with balanced mix of class 0 and class 1, but eventually turn into mostly class 1's?

Sure, you could readily construct a binary time series process that was non-stationary.
Let's make an example. Let $Y_t$ be the random binary value at time $t$ ($t=1,2,...$). Let $S_t$ bet the number of $1$'s observed up to time $t$; i.e. $S_0=0,S_t=\sum_{i=1}^t Y_i\,,\: t = 1,2,...$. 
Now make $p_{t+1}=P(Y_{t+1}=1|S_t)=\frac{1}{1+\exp(-\alpha S_t)}$ for some $\alpha>0$.
So $p_1=0.5$. It will remain at $0.5$ until you observe a $1$, at which point $p$ increases. e.g. if $\alpha=0.3$, then after the first $1$ is observed, $p$ increases to about $0.574$ whereas if $\alpha=0.1$, $p$ only increases to about $0.525$.
(There are many other ways to make nonstationary binary series)
Here's an example using the above scheme with $\alpha=0.005$, for the first 1000 observations. You can see that it starts out with a similar density of 0 and 1 values but the 1-strip rapidly starts to get heavier and the 0-strip gets lighter and lighter, until near the end the 0's have become quite rare.

The conditional probability of a 1 for the last observation ($y_{1000}$) was about 0.979.
A: Yes, they can: For example, take any non-stationary time series and then take its values only at discrete intervals.  That will (usually) give you a non-stationary discrete time series.
