The $\beta$ - function of a coupling determines its energy dependence. This in turn is a function of all the couplings in the theory, usually calculated in perturbation theory. So, things could be complicated for multi-dimensional coupling space.
For a single coupling, assume the one loop result is positive. This means that as long as the coupling is weak, it will grow with the energy scale. If you extrapolate that result way beyond its region of validity, you find that the coupling becomes infinite at some finite energy scale (but, long before that perturbation theory breaks down). This is such a fantastically high energy scale that this so-called Landua pole is an academic issue. Any QFT typically has energy range where it is useful as an effective field theory, and it is not typically valid or useful in such a huge range of energy scales. In any event, at these enormous energy scales quantum gravity is definitely relevant, and it is unlikely to be a quantum field theory at all. For these reasons the Landau pole is no longer a concern for most people, it was more of an issue when QFT was thought to be well-defined at all energy scales.
To your question, since the coupling becomes strong, pretty much anything can happen. It may be that the coupling does diverges at some energy scale (higher or lower than the initial estimate), though to make that statement with confidence you'd need to be able to calculate the $\beta$ - function at strong coupling. If this is the case, your QFT is an effective field theory defined only at sufficiently low energy scales.
It may also be that the $\beta$ - function gets some negative contributions and starts decreasing, whereas a zero becomes possible. When this happens the coupling constant increases initially, but stops running at some specific value. This is the scenario of UV fixed point, which makes the theory well-defined at all energy scales. In this case the problem, such as it is, indeed goes away.
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