Will using teabags in a Keurig cause problems? Am I at risk of damaging my Keurig machine if I place an ordinary black tea bag inside instead of a K-Cup?
It will not damage the machine; However, it won't brew properly.  Your tea is supposed to steep inside the water for a set period of time; but the Keurig will simply pour water quickly through it, resulting in extremely weak tea.

So no, it won't damage the machine, but you will not brew your tea correctly.