Birds? Landing on the sunflowers? I've found that when the flowers are looking like they are ready to set seeds, I put some cheese cloth over them, the birds stay off them and you can get the seeds before they can.

Cuke, look on the back of the head. If you see bite marks or areas that are eaten, I would guess leafhopper/grasshopper.

Cuke,

Make sure you are growing sunflower varieties that are good for cutting. Some varieties have heads the hang down, not matter what, bugs, or no bugs. Unless you are growing them for seed, you should be growing the polenless varieties that don't produce seed. Sunflowers should be picked the morning they open.