There are those (Sephardim and Chassidim, I guess) who have the custom to recite Shir HaShirim on Friday, Erev Shabbat. Do they also recite it on a Friday when Yom Tov leads directly into Shabbat?
If not, is there a reason why?
Otzar Dinim U'Minhagim brings several reasons for saying Shir HaShirim every Friday:
We are preparing ourselves to honor the Shabbat Queen by reading the praises of the bride that are mentioned many times in the Shir HaShirim
Shir HaShirim is a parable of lovers, and Shabbat acts like the broker that connects the Jewish people to their Father in Heaven. So we recite Shir HaShirim then before Shabbat to grow and strengthen the connection between them.
Friday night is the time of intimacy between a man and wife (Eshet Chayil is said for Friday night for the same reason).
The Zohar says that the wicked people in Gehenom are granted respite for the 24 hours of Shabbat, as well as 4 and 1/2 hours every day during prayer (3 prayers, 1 1/2 hours each). This leaves 117 hours a week when the souls are judged in Gehenom. King Shlomo therefore said the 117 verses of Shir HaShirim, in order to be saved from the judgment of Gehinom. Therefore we say it as well in order to be saved from the judgments of Gehenom (this answer is brought in Taami Minhagim #256 as well).
Except for the last reason (117 hours of judgment), all the reasons mentioned would seem to apply both on a regular Friday, and a Friday that was a Yom Tov.
According to the last reason, if a soul is also not judged on Yom Tov (as discussed here), it could perhaps be argued that Shir HaShirim need not be said, since there were not 117 hours of judgment time during that week.