Suppose the order of the finite affine plane is $n\geq 2$ (that is, each line has exactly $n$ points). I claim there are
$$\binom{n^{2}}{4} - (n^{2}+n)\left[\binom{n}{4}+\binom{n}{3}(n^{2}-n)\right]\; (1)$$
such collections.
Assuredly, each line yields:
$\binom{n}{4}$ collections of $4$ collinear points.
$\binom{n}{3}(n^{2}-n)$ collections of $4$ points having $3$ of them collinear. Indeed, each line has $\binom{n}{3}$ collections of $3$ collinear points. But since there are $n^2$ points, for each of these collections we have $n^2-n$ points not on the line to choose from in order to form a collection of $4$ points having $3$ of them collinear.
Hence, each line yields $\binom{n}{4}+\binom{n}{3}(n^{2}-n)$ collections of $4$ points having at least 3 of them collinear. But there are $n^{2}+{n}$ lines and the total number of such collections will then be
$$(n^{2}+n)\left[\binom{n}{4}+\binom{n}{3}(n^{2}-n)\right]\; (2)$$
Finally, since there are $\binom{n^{2}}{4}$ collections of 4 points in total, subtracting $(2)$ from this amount yields $(1)$.
Is this correct? It is in the cases $n=2$ ($1$ collection) and $n=3$ ($54$ collections). Does this type of question fit into a more general setting? That is, well developed counting theories or design theory concepts which answer these types of questions.