Why did he say to bury her because she was a Bas Melech? Her father was a Nochri king. She was also a Yisrael king's wife!
Radak: It is proper to show honor to the esteemed of the nations 1 . She did not deserve honor due to her husband, for she [incited him to sin 2 , and] caused his death and the loss of his kingship.
Rashi (Shemos 6:13, from Pesikta Zutresa Bo 12:31) says that Hashem commanded Moshe to verbally honor Pharaoh.
Melachim I, 21:25. (After seeing the miracle of Har ha'Karmel, Achav let Eliyahu kill Nevi'ei ha'Ba'al. He was ready to repent, but Izevel commanded to kill Eliyahu - Melachim I, 18:40-19:2. - PF)
Why did he command to bury her? Eliyahu's Nevu'ah was that dogs will eat her! We find that Yehu told Bidkar to take Yehoram's body to Navos' field to fulfill the Nevu'ah (verse 25)!
Kli Yakar: Initially he thought that the Nevu'ah was that dogs will eat a small portion of her. When he heard that very little remained, he understood that the Nevu'ah was not only for a little 1 .
Perhaps he assumed that the Nevu'ah was [like verse 36 says] that dogs will eat "Besar (the flesh of) Izavel"; he commanded to bury the bones. However, above (verse 10, and Melachim I, 21:23) it says "Es Izevel." Perhaps "Es" includes even bones, which are not normally eaten. (PF)
Kli Yakar: It is improper to say that he forgot the Nevu'ah, and remembered only after they told him what remained.
