Not in any way that matters. Its an irritant, yes. Can cause short term inflammation. Repeat exposure is, eventually, damaging in measurable ways - like smoking or living in a high smog area long term. Brief, singular exposure isn't something to worry about. Even regular, repeat exposure...
It takes days to work - essentially, it wears away the protective waxes and oils on an insect's carapace, at which point they desiccate - dead of dehydration. Like I said, its a slow kill.
DE also has the unfortunate habit of clumping up and becoming essentially useless in moist environments.
I use it for bug control in my barn, where things stay very dry, and in my plant shed. I do NOT use it in my hen house. More potential problems than its worth. Its also a slow kill, since...
Permethrin. Its a spray. Relatively mild, and quickly broken down by a protein reaction on the skin of all mammals, but absolutely lethal to bugs of all sorts. Ingestion at reasonable dosages will cause no issues whatsoever. No, your chickens can't drink the concentrate, and you should keep...