Answer:
I am wingless but airborne, and when I meet your gaze tears will fall from your eyes. What am I?
John Lennon sang about being this toothy creature.
When they are caught, they are thrown away. When they escape, you itch all day.
To you, rude would I never be, Though I flag my tongue for all to see.
They are producers of pies and burgers and are great for tipping.
I lose my head in the morning and regain back it at night.