I remember in the 40's, 50's and 60's this expression being used instead of "Go to Hell".  It was used to tell someone to get lost, get out of my face, you're lying or simply saying I don't believe a word of what you're saying.

Nowadays folks tend to use more vulgar expressions that imply the same meaning as "Go fly a kite"..

I remember those days. That whole "Go Fly A Kite" thing had negative connotations to me for years. Then, when I got involved with scouting, it went in a different direction. Every year on National Kite Flying Day, we'd get the scouts together and fly kites. It was like a little festival. I was reminded of just how much fun kites can be. So, now when someone tells me to go fly a kite, I might just do that. 

I loved to fly kites...Most of the time I had to make my own.  I took a piece of note book paper, two pieces of wheat straw on each side and for the tail, my mom would tear some pieces of old material to tie on to the bottom....It stayed in the air as long as I could run.