From The game manual:

If the abandoned buildings are not bulldozed, they will be inhabited again after a minimum of 4 weeks of in-game time. Burned down buildings need to be bulldozed to make room for a new building.

So no, you don't have to demolish abandoned buidlings, but demolishing has the following benefits:

1. Avoiding 4-weeks wait.
2. Improving the land value around that location.

You have to demolish burned down buildings, though.

No, demolishing them is not necessary because eventually they will sort themselves out. From The game manual:

If the abandoned buildings are not bulldozed, they will be inhabited again after a minimum of 4 weeks of in-game time. Burned down buildings need to be bulldozed to make room for a new building.

So you don't have to demolish abandoned buidlings, but demolishing has the following benefits:

1. Avoiding 4-weeks wait.
2. Improving the land value around that location.

And you have to demolish burned down buildings, though, they will never recover on their own.