If Eid falls on a Friday, Jumu’ah salat is still obligatory and has to be performed on time, whether one performs Eid salat or not. And this applies for both the Imam and the congregation.
This is the position of the Hanafi, Maliki and Shafi’ and one opinion of Hanbali School of Fiqh.
Ibn Qudamah in his book al-Mughni attributed this opinion to the majority of the Fuqhaa’ and Muslim Jurists.
Their evidence, is the default ruling of Jumu’ah salat as was derived from the ayah in surat al-Jumu’ah:
“O ye who believe! When the call is proclaimed to prayer on Friday (the Day of Assembly), hasten earnestly to the Remembrance of Allah, and leave off business: That is best for you if you knew…”62:9
The binding ruling of Jumu’ah salat was also clearly established by the many ahadith in Bukhari, Muslim and other collections of hadith, and this cannot be out ruled even if Eid falls on Friday.
Jumu’ah salat is also as obligatory as Eid salat, and performing one of them does not substitute for the other. Imam Ibn Hazam rahimahullah in his book al-Muhallaa said: “And if Eid and Falls on Friday, one should pray Eid first and then Jumu’ah. This is a must, and there is no reliable evidence to prove otherwise”. He then said: “Jumu’ah is Fard and Eid is supererogatory, and the supererogatory act does not override the Fard.”
Moreover, the scenario of Eid and Jumu’ah is similar to Eid and Dhuhr salat in any other day besides Friday. Therefore, if dhuhr is required on Eid day, then Jumu’ah which is the Fard of Friday is also required and remains obligatory.
The Hadith some scholars quotes to prove otherwise is the narration of Abi Ubaid, as reported in Muwatta’ Imam Malik, who said: I prayed Eid salat with Uthman ibn Affan one time. He first started with salat, and then delivered the khutbah in which he said: “This day you have two Eids coming in one day, so for the people who came form al-Awaali -the suburbs and outskirts of Madinah- whoever wishes to stay for Jumu’ah, let them stay. And whoever wishes to return home, they can leave. They have my permission.”
However, the scholars mentioned that this was for the people of a specific area, as for them it was not Wajib to Pray Jumu’ah in their locality and it was very hard for them to travel to come back to Madinah.
And Allah knows best