Was incest allowed by God before? Did God allowed incest before? Did Noah and his family committed incest? If yes when did God start forbidding it and why? If no, how did Noah and his family repopulate the world and how about the risk of genetic diseases caused by the inbreeding?
Sexual relations between close relatives was indeed permitted1 until the time of The Law, given through Moses. You can read about it in detail in Leviticus 18 but it is summarised in verse 6:


  No one is to approach any close relative to have sexual relations. I am the Lord.


Why was it forbidden? Possibly because of genetic problems that occur in offspring of close relations, though that is speculation - that chapter does not give any general reason.

1 As an example, Abraham was married to Sarah, his half-sister.

See also:


Was the marriage between Abraham and Sarah incestuous?
Did Adam and Eve's Progeny Commit Incest?