Paying of properties

[Dessa Patton]

I am still a newbie in research mode and I was talking to a family member who has a couple of rental properties. They paid off their properties early and didn't hold the mortgage for the long run. 

Is this a good idea? Didn't they lose tax breaks from the interest that was not paid?

Other than having the property free and clear - and flowing, what other advantages/disadvantages does this situation present?