Is there a source to the fact that Eliyahu Hanavi drinks from the Kos Shel Eliyahu when he comes on the Seder night?
TL;DR the source that Eliyahu "drinks" from each cup at the seder can be (at most) traced back a few hundred years
An OU.org article titled "Eliyahu HaNavi at the Seder" adresses this topic.
Before we determine whether Eliyahu Hanavi drinks from every kiddush cup designated for him at the seder, does he even even APPEAR at the seder?
The article states:
There is no classical (Talmudic or midrashic) source stating that Eliyahu pays a visit to the Seder every year.
A common custom is to take a large decorative cup, fill it with wine, and leave it on the Seder table for a part of, or according to some, for the entire, Seder (see Kitzur Shulchan Aruch 119:1). [This] is known as the Kos shel Eliyahu, Eliyahu’s Cup. This custom is not found in the Talmud, the Rishonim, the Tur, or the Shulchan Aruch.
However, in terms of Eliyahu drinking from the cup:
The lack of a source did not prevent many, in particular Chassidim, from popularizing the idea that Eliyahu makes a furtive appearance, even drinks from the cup, and disappears again.
Perhaps the most extreme proponent of this idea was Rabbi Sholom Dov Ber Schneersohn, the fifth Lubavitcher Rebbe (1860-920) who, before pouring wine from Eliyahu’s Cup back into the wine bottle, would add additional wine because he was concerned that the wine in the cup was pagum (i.e., the cup had been drunk from, rendering the wine unusable for other mitzvot unless additional wine is added) (Rabbi Yehoshua Mundshine, Otzar Minhagei Chabad 175:3, 202).