Do I need to replace the door chime when installing Ring Pro?
A mechanical doorbell chime is a simple electromagnet that energizes when you press the button and magnetically attracts a striker that then contacts a chime plate that makes a ringing sound.  With a higher voltage, the electromagnet will increase in power, and the higher voltage can also cause more heat in the thin wire coil of the magnet.

The higher force could make the chime sound bad, and the extra heat in the wire could break the coil.  I haven't tested this, but if you are going from a 10v transformer to a 16v transformer, I think that the biggest issue will be sound.  I don't think it will burn up the doorbell unless someone is holding down the button, and that's not possible with the Ring, because it simulates the "button push" for a set amount of time.

That said, doorbell chimes are very cheap and unless its very hard to get to, you should just replace the chime.  The old one will most likely sound broken and could actually break down the road.