I've known of people who have ran without drip trays for many years without any side effect. Your engine (especially if your engine is detailed with polished stuff) will get dirty faster and easier with rain, snow, or even washing the car but thats a given. Hell my aftermarket hood is opened up in two sections at least 2" tall and I DD drive my car with no issues.
You'll be fine without the drip trays, and they will disapate heat real well too.
BTW, I was kidding with that comment above! LOL