Shemirath Shabbath, by R' YY Neuwirth (3:58), states very clearly that pouring water over tea leaves resting in a filter that is suspended over a cup, so that water passes through the filter into the cup, is not a problem of Borer. Using water cooled by transferring it to a third vessel (Keli Shelishi) is given there as an acceptable method for brewing fresh tea on Shabbath.
"Tea bags may be used to make tea on Shabbath, but only by putting them into water ... which is already in a keli shelishi..."
"...Boiling water may be poured into a strainer containing tea leaves, upon condition that the tea leaves were boiled up before Shabbath. This does not involve the prohibition against selection, since a) the strainer separates the water from the tea leaves immediately upon its being poured in and b) the water which comes out is the same water which has just been poured in, and it was separate and drinkable even before the whole process took place."
I don't think it is any stretch to apply this to coffee as well.