Can nonstationarity be told from the autocorrelation function? Here "stationarity" means the first and second moments don't change over time.
From a page of Time Series: Theory and Methods, by Peter J. Brockwell, Richard A. Davis

In this chapter we shall examine the problem of selecting an
  appropriate model for a given set of observations $\{X_t t = 1, ..., n
> \}$. If the data (a) exhibits no apparent deviations from stationarity
  and (b) has a rapidly decreasing autocorrelation function, we
  shall seek a suitable ARMA process to represent the mean-corrected
  data. If not, then we shall first look fo r a transformation of the
  data which generates a new series with the properties (a) and (b).

From a page of Time Series: Theory and Methods, by Peter J. Brockwell, Richard A. Davis

Trend and seasonality are usually detected by inspecting the graph of the
  (possibly transformed) series. However they are also characterized by sample
  autocorrelation functions which are slowly decaying and nearly periodic
  respectively. 

From wikipedia

non-stationarity is often indicated by an autocorrelation plot with very slow decay.

My questions:
Why should a stationary ARMA process have a rapidly decreasing autocovariance function? 
Should  a stationary AR process also have a rapidly decreasing autocovariance function? 
Why is non-stationarity often indicated by an autocorrelation plot with very slow decay and values near 1 at small lags?
Why are trend and seasonality characterized by autocorrelation functions that are slowly decaying and nearly periodic, respectively?
 A: (1) A stationary ARMA process has a "rapidly" decaying acf because the acf shows the extent to which previous observations predict the current observation. An AR(1) model would have one significant lag, and AR(2) would have two significant lags....and so on. 
Strictly speaking, an ACF cannot tell you whether the time series is non-stationary. You can no more discern whether or not a series is non-stationary by looking at the ACF than you can by just graphing the series. That is to say, it's just sort of obvious. GDP, for example, obviously exhibits a trend...but you should still use tests - ADF, KPSS, etc. - to properly diagnose the non-stationarity. The slow decaying ACF works the same way. It is not a proper diagnosis per-se, but it is a pretty obvious clue. 
(2) Yes, a stationary ARMA process should have a rapidly decaying ACF. How rapidly depends on the process. Do not assume that just because the ACF doesn't decay after one or two lags that it is non-stationary. You may also have a fractionally integrated series - something you should also test for. 
(3) The ACF looks very slow because effects of previous innovations on a stationary series die out over time. This is why the AR / MA coefficients of less than one are important. You can think of these coefficients as "discount factors" for the influence that previous innovations are having on the current period. They decay because the the structure of the series is such that last period matters the most, two periods ago matters a little less, three periods ago matters less than two periods but is still significant, until you go far enough back that innovations from several periods ago no longer influence current observations. In a non-stationary series, there is no decay.
(4) Slowly decaying is answered above. Seasonality doesn't necessarily imply non-stationarity. It just means that there is some periodicity in the the observations that you must take into account. If you have a monthly consumer spending time series, you might see some seasonality. Every twelve months there is a huge spike in consumer spending around the holidays. That periodicity needs to be taken into account when modeling the time series. 
A: It is possible to get a general formula for stationary ARMA(p,q) autocovariance function. Suppose $X_t$ is a (zero mean) stationary solution of an ARMA(p,q) equation:
$$\phi(B)X_t=\theta(B)Z_t$$
Multiply this equation by $X_{t-h}$, $h>q$, take expectations and you will get
$$r(h)-\phi_1r(h-1)-...-\phi_pr(h-p)=0$$
This is a recursive equation, which has a general solution. If all the roots $\lambda_i$ of polynomial $\phi(z)=1-\phi_1z-...-\phi_pz^p$ are different,
$$r(h)=\sum_{i=1}^pC_i\lambda_i^{-h}$$
where $C_i$ are constants which can be derived from the initial conditions. Since $|\lambda_i|>1$ to ensure stationarity it is very clear why the autocorrelation function (which is autocovariance function scaled by a constant) is decaying rapidly (if $\lambda_i$ are not close to one). 
I've covered the case of unique real roots of the polynomial $\phi(z)$, all other cases are covered in general theory, but formulas are a bit messier. Nevertheless the terms $\lambda^{-h}$ remain.
Answers to question 2 and 3 more or less follow from this formula. For $AR(1)$ process $r(h)=c\phi_1^h$ and when $\phi_1$ is close to one, i.e. close to non-stationarity, you get the behaviour you describle. The same goes for general formula, if the process is nearly unit-root one of the roots $\lambda_i$ is close to 1 and it dominates other terms, producing the slow decay. 
A: As @Clayton pointed out sample autocorrelation function cannot tell if series is non-stationary. 
Problem is that there is various levels of non-stationarity, deterministic OR stochastic trends are just one way time series to be non-stationary. 
Non-constant variance can be also problem even when expected value is independent of time since confidence intervals based on simple sample statistics are invalid. 
