Most high quality interior trim paint can be washed to remove the grunge that accumulates. Polyurethane will hold up to more scrubbing...perhaps for longer than a latex paint, but a high quality acrylic will stand up to repeated cleanings too. Use a cleaning product with surfactants, like 409, to easily clean your white trim. For the deep crevices in white trim, like windows, a toothbrush and 409 is an alternative. A Mr. Clean eraser sponge is also a good product and it will take care of those hand prints on walls with flat paint too.

Personally, I wouldn't. You run the risk of drips. And sometimes poly can give a light brown tint if it gets thick,
not good over white.

what paint are you using on the trim

I wouldn't paint poly over the trim. What trim paint are using?

I poly'ed over white paint before and no problems. I used a brush so Sara is right, watch out for drip marks. But drip marks are avoided simply by not rushing to complete job. Apply thin coats. I didn't see no discoloration when i did it but the poly will change its light reflective properities, which you may or may not like, i.e. do you like your trim in a flat white ? Apply poly in a small area at first as a test study