The network is the function.
A network is a function, that is modeled by terms describing the architecture and coefficients that are learned.
Look at a simple model:
$$f(x) = ax+b$$
Your solver determines $a$ and $b$, and you substitute them into $f(x)$ and then you're able to calculate $f(42)$. The function is linear by definition, but may not be a good fit for your data.
To see if the input data may belong to a linear function can be achieved, when you have a network, that can both fit linear functions and higher order functions. For example try to fit your data to
The function is linear, when $a=0$. If the function is quadratic you get $a\ne 0$ and if it has a higher order than $2$ you will get not good fit with any $a, b, c$.
When looking at MLP and similar networks, the question "is it linear" may not be that easy to answer, but the question how to get the function is the same: Your trained network is the function.
I guess one option to see if the function approximated by an existing network is linear is to train a second network that only contains linear elements. If it is able to approximate your first network, the first network is linear as well. Of course you will need the right test data, as you may accidentally choose a training set that has linearly correlated output in a non-linear function. This is of course the same problem as always, that your network will always can only be as good as the training data.