How can I interpret 不要为你的美丽伤了我?

不要为你的美丽伤害了我

because some people would think wearing a flower would make people looks more pretty. so this "pretty" is the "你的美丽"

you can understand like this:

摘了我(花朵,放到你的头上)确实能让你美丽, 但是请不要为了达到"你的这种美丽的目的" 而做伤害我的事!

Not just "mainland-Chinese public service announcement type signs".

Remember Alice in Wonderland? (Or was it Through the Looking Glass?)

"eat me" "drink me"

Probably Lionel Rowe is right, it's meant to discourage people from taking photos on the grass.

Maybe you grow to a great height if you tread on the grass, or disappear in your camera if you take a photo!

不要为你的美丽伤了我。
Don't damage me for your beauty's sake.

不要践踏草地。

This is quite typical of language in mainland-Chinese public service announcement type signs, which tend to be quite "cutesy" in tone. Stuff like personifying inanimate objects (such as flowers) and writing it from the object's point of view fits with this tone.

The literal meaning is:

don't hurt me for [the sake of] your beauty

Which I suppose is hinting that the reason you might trample the grass is to take an attractive selfie/photo.

Edit: From the photo you attached, it's more likely referring to the flowers than the grass (in other words, don't harm the flowers by picking them).