Would you take off knowing a door is open?
Last week we were waiting in line for 17 in MSP. Ahead of us and #2 for TO was a Mesaba CRJ200 with the high pressure air door hanging open under the left engine. A pinnacle flight radioed "Mesaba xxxx your high pressure air door is open" and Mesaba responded "thanks". They took off soon after.
I don't know what kind of damage that does or what kind of noise it makes in the air but it didn't seem like the best idea to me. I'm really hoping they didn't just check the doors page and figure everything was good.