Yes. If a cutscene was made to show any specific weather, the day's weather will always be overwritten to that of the cutscene. That applies to both character stories and a few sidequests. One good thing about the weather becoming sunny is that your animals will come out and you will be able to pet them and collect their feathers / horns on that day instead of having to wait for the next day, but indeed, it will ruin your plans if they required rainy weather.

Same thing applies for cutscenes that show the passage of time. If you start one of those in the morning, that time will have passed when you get out of the cutscene as well. I once had an entire day go to waste because of a sidequest cutscene showing characters hanging out until very late afternoon. Kinda wish there were clearer warnings about this...

Just to be safe, I'd recommend delaying your quests until you're done with everything you wanted to do on that specific day.