The correct scholarly view is that whoever eats thinking that Fajr (dawn) has not yet come, then finds out that it had in fact come, does not have to do anything, because he was unaware of the time, so he is excused.
Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) said: If a fasting person eats or drinks anything that breaks the fast out of ignorance, then his fast is still valid, whether he was unaware of the time, or unaware of the ruling, such as a man who gets up at the end of the night, and thinks that dawn has not yet come, so he eats and drinks, then he finds out that dawn had in fact come – his fast is still valid because he was unaware of the time. An example of one who is not aware of the ruling is a fasting person who has cupping done and does not know that cupping invalidates the fast, so we say to him: your fast is still valid. The evidence for that is the words of Allaah (interpretation of the meaning):
“Our Lord! Punish us not if we forget or fall into error, our Lord! Lay not on us a burden like that which You did lay on those before us (Jews and Christians); our Lord! Put not on us a burden greater than we have strength to bear. Pardon us and grant us forgiveness. Have mercy on us. You are our Mawlaa (Patron, Supporter and Protector) and give us victory over the disbelieving people”
That is the evidence from the Qur’aan. The evidence from the Sunnah is the hadeeth of Asma’ bint Abi Bakr (may Allaah be pleased with her) which was narrated by al-Bukhaari in his Saheeh. She said: “We broke our fast on a cloudy day at the time of the Messenger of Allaah (peace and blessings of Allaah be upon him), then the sun came out.” That meant that they had broken their fast during the day, but they did not realize, because they thought that the sun had set. But the Prophet (peace and blessings of Allaah be upon him) did not tell them to make up that day; if it had been obligatory to make it up he would have told them to do so. And if he had told them to do that it would have been transmitted to us.
Majmoo’ al-Fataawa, 19.
See also question no. 38543.