We often hear this story from the pulpits and also from lecturers and preachers.
The summary of the widespread story amongst the people is:
“Umar (may Allaah be pleased upon him) was sitting with some of his companions. He suddenly laughed a little and then started crying. When the people who were with him asked him why he did this, he said:
When we were in Jaahiliyyah, we used to make an idol out of ‘Ajwa dates, we used to worship it then eat it. This is what made me laugh.
As for me crying, I used to have a daughter and I wanted to bury her alive so I took her with me and I dug a hole for her, she then began to play with my beard so I buried her whilst she was alive.
How strange that you find a person amongst the masses, he does not comprehend anything regarding Islaam, neither does he memorize anything regarding its rulings but he knows the story of Umar burying his daughter alive!
The sad reality is that the preachers and lecturers will bear the sin of spreading this fabricated story which has been falsely attributed to the Chief of the Believers Umar Ibn al-Khattaab (may Allaah be pleased with him).
Strangely, this story has no mention in the books of Sunnah, Hadeeth, nor the books of narrations or history. Also nothing is known about the origins of the story except that the Raafidhah (Shee’ah) have fabricated it, out of jealousy, and without any evidence or proof.
This type of narration cannot be deemed accurate unless it has an authentic chain of narration, and we do not have an authentic chain of narration proving that Umar (may Allaah be pleased with him) did such an action.
Additionally, there are other facts that prove this story is fabricated,
1. It is known that the first woman Umar married was Zaynab bint Madh’oon (may Allaah be pleased with her), the sister of Uthmaan. The children born to them were: Hafsah, Abdullah and AbdurRahmaan al-Akbar. 
The birth of Hafsa was five years before prophethood, so she was the oldest daughter of Umar. 
So why did Umar Ibn Al-Khattaab (may Allah be pleased with him) not bury his daughter Hafsa (may Allaah be pleased with her) and she is his oldest daughter?
In fact his Kunya is Abu Hafsah
Why would he bury his daughter who is younger than her?
Why is there no information regarding the daughter that he supposedly buried? None of his relative mentioned this story, neither is she mentioned along with his other children. 
2. It was not known about the tribe of Banee Adiyy, to whom Umar Al-Farooq ascribed to, that they used to bury their daughters alive. This is shown by the fact that his own sister Faatimah, remained alive and married their cousin Sa’eed Ibn Zayd.
3. After searching the books of Hadeeth and Takhreej (wherein the books of Hadeeth are analysed and graded), I did not find a single narration except in the books of the Raafidhah (Shee’ah). It is not mentioned in the books of Sunnah or Hadeeth, and neither is it mentioned in the books of Aathaar (narrations) and History. This in itself is a clear indication that the story is absolute falsehood..
3. Finally, one of the tactics used in insulting the companions is to broadcast stories which are “innocent” if taken in their apparent sense. Therefore, it is befitting for each person to verify and know where he is taking knowledge from.
So this story is from the fabrication and lies of the Rawaafidh.
The one who is accused of fabricating this story is the corrupt Raafidhee Ni’matullah Al-Jazaairee, the one who wrote Al-Anwaar an-Nu’maaniyyah. In order to hide the fact that the story is fabricated, he used the term: ‘it has been narrated…’ [4 ]
We should all spread this message in order to expose this lie, and in order to defend Chief of the Believers Umar Ibn Al-Khattaab and to defend all the companions of the Prophet (salaam Allaahu alayhi wa salaam).
 This has been mentioned by Ibn Katheer in Al-Bidaayah wan-Nihaayah. He said, “Al-Waaqidee, Ibn Al-Kalbee and others said that Umar married Zaynab bint Madh’oon in Jaahiliyyah, the sister of Uthmaan Ibn Madh’oon. The children born to them were: Hafsa, Abdullah and AbdurRahmaan al-Akbar.”
 As has been narrated by Al-Haakim in al-Mustadrak and others: ‘Umar said: “Hafsa was born whilst the Quraysh were building the Ka’bah. This was before the prophethood by five years.”
 The biography of the mother of the believers, Hafsah (may allaah be pleased with her) in Al-Isaabah by Al-Haafidh Ibn Hair 7/582
[4 ] i.e. he did not say: “such a person said or heard.” This type of wording would give us a level of confidence that the story has been narrated first hand. However when it is said, ‘is has been narrated,’ it shows that the narration may not be accurate.
Share with every Muslim
Answer by Moulana Mufti Muntasir Uz Zamaan.
Graduate of Darul Uloom Azaadville and Darul Iftaa Sherwood.
Also see – http://hadithanswers.com/did-sayyiduna-umar-radiyallahuanhu-bury-his-daughter-alive-before-islam/