Merike Ri Berit reads in part “When Merieke Ri Berit leaves the battlefield or becomes untapped, destroy that creature. It can't be regenerated.” As I read the oracle text this seems to be a part of the control gaining effect, not a stand alone trigger. I.e. this is like the delayed triggered ability of Necropotence, and it does not matter if Necro is gone or if Merieke loses this ability prior being untapped - The effect will still happen.

The interaction of interest is with Song of the Dryads enchanting Merieke.

Question:
Will Merike untap after being turned into a forest by Song of the Dryads, and will the creature she controls be destroyed at that moment?

Yes, the trigger you quoted is a delayed triggered ability set up during the resolution of Merieke Ri Berit's control-gaining ability, not a separate, independent trigger. As such, when a Merieke Ri Berit enchanted by Song of the Dryads untaps (as it will during your untap step, since it no longer has an ability that specifies it does not), the delayed trigger will go off and destroy the creature you gained control of.