No, FF14 doesn't seem to have any kind of borrowed power systems.
PotD, HoH, Eureka, and Bozja are the "closest" it has. However the major difference is that the power gains are permanent (not removed once the next patch or expansion comes out), and they only apply to optional content which you can completely ignore. For instance, I have barely touched HoH, yet my character is no worse off than someone who spends every waking moment in there.

A true borrowed power system would have been if they made PotD's aetherpool levels apply in all content, thereby forcing you to do PotD, and if the power boost was taken away once the next expansion dropped, nullifying all your work (this is the "borrowed" part). Basically, WoW's artifact weapon system, heart of azeroth levels, and now covenant ranks. Players are forced to engage with those systems to keep their characters viable, and all their hard work advancing those systems will be completely wiped away once WoW's next expansion drops.