The Mishna (Pesahim 10:8) states that one may not partake of desserts or similar foods after eating the Korban Pesah, which is now symbolized by the Afikoman, and this is codified in the Shulhan Aruch (Orah Haim 478:1). The Rambam (Hilchot Hametz Umatza 8:9) explains that this is so that the taste of the Korban Pesah (or nowadays, the Matza) remains in one’s mouth after the Seder.
The Mishna Berura (O.H. 478:2) explains that there is a debate among the Poskim as to whether drinking beverages after the Afikoman falls into the same restriction as food, and concludes that one should be strict and only drink water or similar liquids if needed. Similarly, the Shulhan Aruch (Orah Haim 481:1) writes that after the four cups of wine, one should not drink anything, except water. Nevertheless, the following Halacha (Orah Haim 481:2) states that after the Seder, one should remain awake to learn the laws of Pesah and to further delve into the Exodus from Egypt. As such, it would appear to be a challenge to learn after a long Seder while not being able to drink coffee or tea, which could aid in keeping one awake and focused.
The HIDA (Birke Yosef, § 481 & More BaEtzba, § 211) writes that the custom is to permit coffee or tea after the Afikoman, but that sugar should not be added. Rabbi Yosef Shalom Elyashiv (Hagada) writes that one may use sugar. The Mishna Berura (ibid.) says that one should not have any intoxicating beverages and that in addition to water, one may have similar drinks, which nowadays would apply to juices or soft drinks.
Summary: One may drink water, coffee, tea, juices or soft drinks after the Afikoman and the four cups of wine. One should learn about Pesah after the Seder until one becomes sleepy.