Why does a perturbative theory imply coupling must be increasing or decreasing?

I'd agree that the sentence is not correct (in a sense that the author should't have written "must"). There are actually three possibilities:

First:

If the coupling increases with $\mu$, as in QED, it goes to zero at long distances.

Second:

If it decreases with $\mu$, as in QCD, it goes to zero at sort distances...

And the third possibility:

The third possibility in a perturbative theory is that $\beta(\alpha) = 0$ exactly, in which case the theory is scale-invariant.

Also I'd agree that it has nothing to do with perturbativity at all. Since even if the theory is non-perturbative...

Even if the theory is non-perturbative, one can still define a coupling through the value of the Green's function.

BTW -- all the quotes above are from the same paragraph where your sentence is coming from. So I'd say that that sentence is quite unfortunate.

Edit: As for the questions in comments -- I'll add just another quote:

With multiple couplings there are other possibilities for solutions to the RGEs. For example, one can imagine a situation in which couplings circle around each other. It is certainly easy to write down coupled differential equations with bizarre solutions...

So, yes, there are possibilities for more complex RG behaviors. I guess what author wanted to do -- is to give an overview that goes from simple examples to more complex ones. Starting with perturbative single-coupling QFT with $\beta \ne 0$ and then adding more complex possibilities. And I'd agree that the whole paragraph is not very good at conveying this idea.

Perturbation theory and the fact that couplings run in most quantum field theories are not actually directly related to each other. As long as the coupling is small enough, one can use perturbation theory. The smaller the better of course.

The coupling runs, because the strength of the interaction differs at different energy scale. This shows up when one computes higher orders in the perturbative expansion. From the results of the higher order calculations one can compute something called the beta-function and from this one can compute how the coupling runs. For more detail on how this works, one can read up on renormalization group analysis.

It may happen (as it does in quantum chromodynamics or QCD) that the coupling becomes so strong at a particular energy scale that perturbation theory breaks down. The perturbative expansion does not work in this situation, because higher orders become as significant as, if not more significant than, the lower orders. In such situations, one needs to resort to so-called non-pertubative methods (such as Schwinger-Dyson equations).