Is there an issue of borer when using the plunger of a French Press?
What are the possible issues involved?
Is it borer with a kli?
Of course CYLOR, but the short answer is yes.
The longer answer is look in Hilchos Yom Tov siman 510 siff two and three with the Mishna Berurah and Biur Halacha.
Basically any Borer you can't do before Yom Tov due, for quality purposes, is allowed to be done on Yom Tov. So coffee which is only good fresh is allowed to be prepared in a way that necessitates Borer on Yom Tov.
There is another opinion dealt with in the Biur Halacha where an item which is sometimes separated for many days use should not be done on Yom Tov. This concern does not apply to the amount of coffee in a French Press.
This is all true even when doing the type of Borer which is most stringent Halakhicaly which is psoless from ochel, taking bad from good. A French Press would have an additional leniency of being ochel from psoles, taking good from bad, being that pressing down the plunger is not Borer as all products are still mixed. Its only when pouring that the liquid seperates from the grounds.