Sorry for the clickbait title. No, really, I feel bad.
In any case, here are some cool little things you can do to your paintings. Okay, I know, they're super gimmicky. I'm not advocating doing this to every piece in your portfolio. It's just - you know - cute. Used in moderation, it can be very effective, like the F-word - or salt. Cue, Buzzfeed writing.
Everybody loves a good Depthy. All you need is a flat .jpeg and a depth map of it. Those are really easy to make especially for paintings since you already have all your layers in place.
2. After Effects Parallax
This one's a bit harder to pull off but the pay off is much more satisfying, I feel. However, you need access to After Effects and need to spend a little more time learning the basics of the program. Having said that, it really is all you need to know about After Effects. The basics. Five minutes on Youtube and you're good. Once again, its much easier for paintings since all your objects have already been isolated into layers.
Keep it simple, keep it subtle.