Rashi (Daniel 1:1): It was the third year of his rebellion. Malbim - : It is called the third year, for it is the third year that he ruled by himself. We say similarly about Hoshe'a ben Elah and Uziyah. If so, "Kevuras Chamor Yikaver Sachov v'Hashlech me'Hal'ah l'Sha'arei Yerushalayim" (Yirmeyah 22:19) means that they sent his corpse from Bavel to Eretz Yisrael, to drag it in disgrace around the gate of Yerushalayim. Alternatively, when Nebuchadnetzar came against Yehoyachin, he brought Yehoyakim with him, and there he was killed and dragged to disgrace his son who ruled in place of him.

Malbim: However, in Yirmeyah (36:9, 23) it says that in Yehoyakim's fifth year they called a fast, and Yehoyakim burned Megilas Eichah that Yirmeyah wrote. This teaches that in his fifth year, he was in Yerushalayim! Do not say that this was after he returned to Yerushalayim, and he was a slave to Nebuchadnetzar. When he burned the Megilah, he said "why did you write that Melech Bavel will come