Grading: Its chain is "extremely weak" (da'if jiddan).
Technical note / Defect ('illah): Muhammad bin al-Zubayr al-Hanzali al-Basri is a "matruk" (abandoned) narrator. Imam al-Nasa'i said in "al-Mujtaba" that Muhammad bin al-Zubayr is da'if and the likes of him do not establish a proof, and that there is also disagreement over him in the narration of this hadith, as the author (al-Hakim) will mention in the following narrations. He also said that the other hadith likewise revolves around Muhammad bin al-Zubayr al-Hanzali and it is not sahih. Moreover, al-Hasan al-Basri's hearing from 'Imran (may Allah be pleased with him) is not established. Sufyan mentioned in the chain refers to Sufyan al-Thawri.

Reference / Source: Imam Ahmad (33/19985) narrated it via 'Abd Allah bin al-Walid, and al-Nasa'i in "al-Mujtaba" (3847) through 'Umar bin Abi Zayd Abu Dawud al-Hafari, and both of them narrate it from Sufyan al-Thawri with this same chain.

Details of the narration: In Musnad Ahmad its wording is: "There is no vow in disobedience to Allah or in anger", and in Sunan al-Nasa'i the wording is: "There is no vow in disobedience, nor in anger."

Reference / Source: Imam Ahmad (19945) and al-Nasa'i (3848) narrated it through Abu Bakr al-Nahshali with the chain Muhammad bin al-Zubayr from al-Hasan al-Basri; however, in al-Nasa'i's narration only the word "disobedience" occurs, and "anger" is not mentioned.

Reference / Source: Imam Ahmad (19856), Imam al-Nasa'i in "al-Kubra" (8709) and "al-Mujtaba" (3849), and Imam Ibn Hibban (4392) narrated it as part of a story, through Mansur bin Zadhan with the chain al-Hasan al-Basri from 'Imran bin Husayn.
Details of the narration: The Prophet (peace and blessings be upon him) said: "There is no vow for the son of Adam in what he does not own, nor any vow in disobedience to Allah."
Grading: With this wording the hadith is "sahih", and this chain is more sound (than the other chains).
Technical note / Defect ('illah): Although Imam al-Hasan al-Basri did not hear directly from 'Imran bin Husayn (may Allah be pleased with him), he has been corroborated (mutaba'ah), as will be detailed in the takhrij of narration no. (8039).

Note / Clarification: As for the issue that the expiation for a vow is the same as the expiation for an oath, this has been narrated from a group of the Companions.
Reference / Source: In this connection, see the hadith of 'Uqbah bin 'Amir (may Allah be pleased with him) in "Musnad Ahmad" (no. 17301); we have mentioned its other supporting reports there in detail.