It is obligatory to straighten the rows and make them complete (by filling any gaps) when praying

It is obligatory to make the rows straight in prayer. That has been discussed previously in the answer to question no. 36881. There is no difference in this regard between Eid prayer and other prayers. 

Part of this straightening is completing the rows and connecting them, and not leaving gaps between worshippers. Abu Dawood (666) narrated from ‘Abdullah ibn ‘Umar (may Allah be pleased with him) that the Messenger of Allah (blessings and peace of Allah be upon him) said: “Straighten the rows, stand shoulder to shoulder and fill the gaps; be gentle with their hands of your brothers, and do not leave gaps for the Shaytaan. Whoever connects a row, Allah will connect him (with His mercy), and whoever breaks a row, Allah will cut him off (from His mercy).”

Classed as saheeh by al-Albaani 

Muslim (430) narrated that Jaabir ibn Samurah (may Allah be pleased with him) said: The Messenger of Allah (blessings and peace of Allah be upon him) came out to us and said: “Why do you not make your rows as the angels make their rows in the presence of their Lord?” We said: O Messenger of Allah, how do the angels make their rows in the presence of their Lord? He said: “They complete the first rows and they make the rows compact.” 

Abu Dawood (671) and an-Nasaa’i (818) narrated from Anas ibn Maalik (may Allah be pleased with him) that the Messenger of Allah (blessings and peace of Allah be upon him) said: “Fill up the first rows, then the next ones, and if there is any deficiency, let it be in the last row.”

Classed as saheeh by al-Albaani in Saheeh Abi Dawood and elsewhere. 

Al-Baaji (may Allah have mercy on him) said:

It is obligatory to complete the first row, then the next, and if there is any deficiency in should be in the last row.

End quote from al-Muntaqa Sharh al-Muwatta’ (1/386) 

It says in al-Mawsoo‘ah al-Fiqhiyyah (27/36): 

Part of straightening the rows is completing the first row, then the next row; it is not prescribed to start the second row until after the first row has been completed, and so on. This is a matter concerning which the fuqaha’ are agreed. 

Based on that, no one should stand in a row when in front of him there is another row that is deficient or in which there is a gap; rather he should push through the rows to fill a gap in the rows in front of him. End quote. 

Shaykh Ibn Baaz (may Allah have mercy on him) was asked: 

If there is a gap in a row during Taraweeh or qiyaam because some of the worshippers have left, should the imam ask those in the second row to complete the first row? 

He replied: 

It is obligatory for the members of the congregation in an obligatory or naafil prayer to complete the first row, then the next, because the Prophet (blessings and peace of Allah be upon him) enjoined and encouraged that, as he (blessings and peace of Allah be upon him) said: “Make your rows straight and fill the gaps.” And he (blessings and peace of Allah be upon him) said: “Why do you not make your rows as the angels make their rows in the presence of their Lord?” They said: O Messenger of Allah, how do the angels make their rows in the presence of their Lord? He (blessings and peace of Allah be upon him) said: “They complete the first rows and they make the rows compact.” 

End quote from Majmoo‘ Fataawa Ibn Baaz (30/124-125) 

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said: 

It is prescribed for the imams to ask the people to straighten the rows in Eid prayer and prayer for rain (istisqa’) as it is in other prayers, because if the people are not reminded of that, they may become heedless concerning it. It is prescribed in every congregational prayer, if the people are forming rows, to remind them and tell them to make their rows straight. End quote. 

Liqa’ al-Baab al-Maftooh (46/9) 

He also said: 

Many imams and many worshippers are careless about the matter of making the rows compact, so you will find a row in which there are many gaps that no one fills. This is a mistake, because the Prophet (blessings and peace of Allah be upon him) instructed the people to make the rows compact, and said that the angels before Allah, may He be glorified and exalted, make their rows compact. End quote.

Fataawa Noor ‘ala ad-Darb by Ibn ‘Uthaymeen (111/161) 

If the people pray with a gap in the row, they have done wrong, but their prayer is still valid. 

Al-Haafiz Ibn Hajar (may Allah have mercy on him) said: 

Although we say that straightening the rows is obligatory, the prayer of one who does not do that and does not straighten the row is still valid. End quote. 

Fath al-Baari (2/210) 

And Allah knows best.