Is ribbon bad for birds?

My lovebird loves shredding things. She recently has got her eye on some ribbons (the type you tie party balloons with) I have lying around. I would love to tie some of her toys with them in her cage, but she tends to eat most things she shreds. I want to know how harmful it could be to her. Also, when she's out of her cage, she does the same with fabric, is this dangerous too?