What should I do about putty that fell in the toilet opening?
Try a bent wire coat hanger to pierce and hook the putty. If you can't, does water still seem to drain? I would pour some water down there slowly to see if it's draining properly. You could even try boiling water which might make the putty more malleable and help flush it down.

Ultimately, if it's draining I don't think you need to worry about it. Just think about all the other things that get flushed down the toilet.