Are the Old Testament prohibitions on incest applicable to Christians? The Old Testament forbids incest. Does this aspect of the Mosaic law apply to Christians?
Are the Old Testament prohibitions on incest applicable to Christians? - Yes.
Leviticus 18:16 is observed in the Gospel of [Mark 6:17-18], alluding to Christian observance all the Laws in Leviticus 18 regarding incest.

Leviticus 18:16 in regards to Mark 6:17-18 :
Mark 6:17-18 [KJV]
17 For Herod himself had sent forth and laid hold upon John, and bound him in prison for Herodias' sake, his brother Philip's wife: for he had married her.
18 For John had said unto Herod, It is not lawful for thee to have thy brother's wife.


Leviticus 18:16 [MT] 
You shall not uncover the nakedness of your brother's wife. It is your brother's nakedness. (עֶרְוַ֥ת אֵֽשֶׁת־אָחִ֖יךָ לֹ֣א תְגַלֵּ֑ה עֶרְוַ֥ת אָחִ֖יךָ הִֽוא)