The model building process involves a model builder making many decisions. One of the decisions involves choosing among different classes of models to explore. There are many classes of models that could be considered; for example, ARIMA models, ARDL models, Multiple Source of Error State-Space models, LSTAR models, Min-Max models, to name but a few. Of course, some classes of models are broader than others and it's not common to find that some classes of models are sub-classes of others.
Given the nature of the question, we can focus mainly on just two classes of models; linear models and non-linear models.
With the above picture in mind, I'll begin to address the OPs question of when it is useful to adopt a non-linear model and if there is a logical framework for doing so - from a statistical and methodological perspective.
The first thing to notice is that linear models are a small subclass of non-linear models. In other words, linear models are special cases of non-linear models. There are some exceptions to that statement, but, for present purposes, we won't lose much by accepting it to simplify matters.
Typically, a model builder will select a class of models and proceed to choose a model from within that particular class by employing some methodology. A simple example is when one decides to model a time-series as an ARIMA process and then follows the Box-Jenkins methodology to select a model from among the class of ARIMA models. Working in this fashion, with methodologies associated with families of models, is a matter of practical necessity.
A consequence of deciding to build a non-linear model is that the model selection problem becomes much greater (more models must be considered and more decisions are faced) when compared to choosing from among the smaller set of linear models, so there is a real practical issue at hand. Furthermore, there may not even be fully developed methodologies (known, accepted, understood, easy to communicate) to use in order to select from some families of non-linear models. Further still, another disadvantage of building non-linear models is that linear models are easier to use and their probabilistic properties are better known (Teräsvirta, Tjøstheim, and Granger (2010)).
That said, the OP asks for statistical grounds for guiding the decision rather than practical or domain theoretic ones, so I must carry on.
Before even contemplating how to deal with selecting which non-linear models to work with, one must decide initially whether to work with linear models or non-linear models, instead. A decision! How to make this choice?
By appeal to Granger and Terasvirta (1993), I adopt the following argument, which has two main points in response to the following two questions.
Q: When is it useful to build a non-linear model? In short, it may be useful to build a non-linear model when the class of linear models has already been considered and deemed insufficient to characterize the relationship under inspection. This non-linear modelling procedure (decision making process) can be said to go from simple to general, in the sense that it goes from linear to non-linear.
Q: Are there statistical grounds that can be used to justify building a non-linear model? If one decides to build a non-linear model based on the results of linearity tests, I would say, yes, there are. If linearity testing suggests that there is no significant nonlinearity in the relationship then building a nonlinear model would not be recommended; testing should precede the decision to build.
I will flesh these points out by direct reference to Granger and Terasvirta (1993):
Before building a nonlinear model it is advisable to find out if
indeed a linear model would adequately characterize the [economic]
relationships under analysis. If this were the case, there would be
more statistical theory available for building a reasonable model than
if a nonlinear model were appropriate. Furthermore, obtaining optimal
forecasts for more than one period ahead would be much simpler if the
model were linear. It may happen, at least when the time-series are
short, that the investigator successfully estimates a nonlinear model
although the true relationship between the variables is linear. The
danger of unnecessarily complicating the model-building is therefore
real, but can be diminished by linearity testing.
In the more recent book, Teräsvirta, Tjøstheim, and Granger (2010), the same sort of advice is given, which I now quote:
From the practical point of view it is [therefore] useful to test
linearity before attempting estimation of the more complicated
nonlinear model. In many cases, testing is even necessary from a
statistical point of view. A number of popular nonlinear models are
not identified under linearity. If the true model that generated the
data is linear and the nonlinear model one is interested in nests this
linear model, the parameters of the nonlinear model cannot be
estimated consistently. Thus linearity testing has to precede any
nonlinear modelling and estimation.
Let me end with an example.
In the context of modelling business cycles, a practical example of using statistical grounds to justify building a non-linear model may be as follows. Since linear univariate or vector autoregressive models are unable to generate asymmetrical cyclical time-series, a non-linear modelling approach, which can handle asymmetries in the data, is worth consideration. An expanded version of this example about data reversibility can be found in Tong (1993).
Apologies if I've concentrated too much on time-series models. I'm sure, however, that some of the ideas are applicable in other settings, too.