For the model $X_t = \sum_{j=-\infty} ^{\infty} \psi_j Z_{t-j}$, where $Z_t \sim WN(0, \sigma^2)$, I'm not totally clear on why we require $\sum_{j=1}^{\infty} | \psi_j| < \infty$.
I think we can show that $$ E \left[\sum_{j=-\infty} ^{\infty} |\psi_j Z_{t-j}| \right] \le \sigma \sum_{j=-\infty} ^{\infty} |\psi_j|$$ so the left hand side converges if the right hand side does. But why is it important for the left hand side to converge?
Short Answer: Requiring $\sum_j |\psi_j| < \infty$ avoids a few strange behaviors easier without being a much stronger assumption, so folks make it to avoid having to caveat all their other theorems.
Longer:
When building a model, you're almost certainly going to want a finite variance (unless you're specifically building a 'heavy-tailed' model). For this, we only need the slightly weaker condition that $\sum_j \psi_j^2 < \infty$. [SS15, Definition 1.12]
$$ \begin{align*} \text{Var}\left[\sum_{j=1}^{\infty} \psi_j Z_j\right] &= \sum_{j=1}^{\infty} \text{Var}[\psi_j Z_j] \\ &= \sum_{j=1}^{\infty} \psi_j^2 \text{Var}[Z_j] \\ &= \sigma^2 \sum_{j=1}^{\infty} \psi_j^2 \end{align*} $$
If the variance exists (is finite), it's a standard result that the mean exists (is finite) as well [S03, Section 1.3.2]. However, if we don't require absolute convergence on the $\psi_j$, the series $\sum_j \psi_j$ may only be conditionally convergent and not absolutely convergent, which leads to strange things like the Riemann Rearrangement Theorem applying. In practice, it's not the RRT that you're worried about - that's just an example of the strange properties of conditionally convergent series. One of the great things about absolute convergence is that it lets you switch around integrals (expectations) and sums: this lets us assume that the sum gives a sensible random variable.
Another, more serious, issue is that, without assuming absolute summability, you can't prove the ergodicity of the mean of the series (ergodicity means that given a long enough observation, you can get a good estimate of the mean which is useful because we typically only have one realization of a time series). The series may be 'long-memory' (long-range dependence) and having more observations won't necessarily make the variance of your mean estimate decay: roughly, any shocks will 'stick around' forever and pollute your estimate of the mean. (See [SS15, Section 5.2]; I also like [S06] for a more general overview of long-memory processes, but it's not the easiest read just because the subject is hard.)
Hamilton [H94] discusses this briefly in section 3.3, particularly footnote 3, where he refers the reader to [R73, p.111] for details, and appendix 3.A but I don't have the Rao reference handy.
[H94] James D. Hamilton, Time Series Analysis, 1st Ed. (1994) Princeton University Press.
[R73] C. Radhakrishna Rao, Linear Statistical Inference and Its Applications 2nd Ed. (1973) Wiley.
[S03] Jun Shao, Mathematical Statistics, 2nd Ed. (2003) Springer. Springer Texts in Statistics.
[S06] Gennady Samorodnitsky, "Long Range Dependence". Foundations and Trends in Stochastic Systems 1(3). p.163-257 (2006).
[SS15] Robert H. Shumway and David S. Stoffer, Time Series Analysis and Its Applications, 3rd Ed. Blue Printing (2015-12). Springer. Freely available at http://www.stat.pitt.edu/stoffer/tsa3/
Since you have an infinite series $\sum_j \psi_j Z_{t-j}$ it is not immediately given that it sums up to a random variable. Furthermore since it is a sum of random variables there are various notions how to understand the sum of such series.
The most simple is the convergence almost surely and for that we have the following statement. If $\sum_j E|Y_j|<\infty$ then the series $\sum_j Y_j$ converges absolutely almost surely and $E\sum_j Y_j = \sum EY_j$.
Since $E|Z_{t}|=const$ for all $t$, if $Z_t\sim WN(0,\sigma^2)$, the condition $\sum_j |\psi_j|<\infty$ ensures that series $\sum_j \psi_j Z_{t-j}$ are well defined, i.e. that $X_t$ exists for all $t$.
