They're rarely used in a continuous tense because it rarely makes sense to do so - we usually think of verbs like start or finish as describing a single moment in time rather than a process. It wasn't raining, then it started raining, then it was raining.
But if you want to emphasize that the starting or finishing took a certain amount of time - to impart a continuous aspect to it - there's nothing stopping you from doing so. For example:
I was finishing my work when John called.
Here, you want to emphasize that you were in the process of finishing your work, and that it took an amount of time, partway through which John called (as a singular, "momentary" action). On the other hand, if you say:
I finished my work when John called.
you describe two of such "momentary" actions, and it'll be interpreted as a sequence of events - you finished your work (as a single action), and then John called.
In your example, it started to rain would evoke a more sudden change - at one moment it wasn't raining, at the next one it was. It was starting to rain would make it feel like more of a process - from the first few drops, to a light drizzle, to a downpour.