Autocorrelation vs Non-stationary What is the relationship between autocorrelation and non-stationary?
Is it true that non-zero autocorrelation $\implies$ non-stationary, but not vice versa?
 A: Autocorrelation doesn't cause non-stationarity. Non-stationarity doesn't require autocorrelation. I won't say they're not related, but they're not related the way you stated.
For instance, AR(1) process is autocorrelated, but it's stationary:
$$x_t=c+\phi_1 x_{t-1}+\varepsilon_t$$ $$\varepsilon_t\sim\mathcal{N}(0,\sigma)$$
You can see that the unconditional mean is $E[x_t]=\frac{c}{1-\phi_1}$, i.e. stationary.
I(1) is non-stationary: $$x_t=x_{t-1}+\varepsilon_t$$ 
Here, errors $\varepsilon_t$ are not autocorrelated, but $x_t$ are autocorrelated: $\operatorname{cov}[x_t,x_{t-1}]\ne 0$. 
Obviously, we want $|\phi_1|<1$, otherwise, AR(1) will blow up, as Richard Hardy noted.
A: Neither is true. Consider the two following examples:
(1) Let $\xi \sim N(0,1)$ and define the stochastic process $X_i=\xi, i=1,2,\dots$. it's easy to check that this process is (strongly) stationary, while at the same time $\mathrm{Cov}(X_i, X_j)=1, \forall i,j$. 
(2) Consider a stochastic process consisting of independent Gaussian variables $X_i \sim N(0, i), i=1,2,\dots$. This process is clearly not stationary, but the autocorrelation is zero for all lags since the variables are independent.
