What happened to Yosef's coat of many colors? I know the brothers gave it to Yakov. But what happened later? Did he have it in Egypt? Do we know?
Presumably, it was buried, because there is a Chiyuv D'Oraisa (biblical obligation) to bury the dead. It is a question in the Poskim of how much remains of the dead requires burial, but a significant amount certainly requires burial.
Yaakov was under the impression that the Kesonos Pasim had absorbed a lot of blood from the dead Yosef. That being case, he would have buried it in fulfillment of the Halacha of burying the dead.
The Talmud says "Romi has another festival: once in 70 years, they put a healthy man (to represent Esav) on top of a lame man (Yakov); 'Esav' wears the clothes of Adam ha'Rishon, has the skin of R. Yishmael over his face, and the weight of a Zuz of Paz (a very rare type of gold) hanging from his neck;"
Rashi there comments about the clothes - והן בגדי חמודות שהיו לעשו
There is an opinion that says this garment Yaakov wore to receive the blessing from his father and passed it down to Yosef.