If a person breaks his fast on a day when he was making up a missed fast, does he have to fast three days?

Making up missed Ramadan fasts is a kind of obligatory fast which it is not permissible to break without a legitimate shar’i excuse. If a person starts to observe a fast to make up for a missed fast, then he is obliged to complete it. 

See also questions no. 39752 and 39991. 

If a person breaks a fast that he is observing to make up for a missed Ramadan fast, he has to make up that day. If he broke that fast with no excuse, then as well as making it up he has to repent to Allah from this sin. 

With regard to what you mentioned about fasting three days to replace that day, there is no basis for this. 

Rather some of the scholars said that he has to fast two days: the day from Ramadan and the day he was making up. 

The correct view is that he only has to make up one day. 

Ibn Hazm said in al-Muhalla (6/271): Whoever breaks his fast deliberately when making up a missed Ramadan fast only has to fast one day, because the obligation to make up the replacement day is not something that was enjoined by Allah. It was narrated in a saheeh report that the Prophet (peace and blessings of Allah be upon him) made up that day of Ramadan so it is not permissible to add anything else to that without any textual evidence or scholarly consensus. 

The Shaykh (may Allah have mercy on him) was also asked: In the past I fasted some days to make up what I owed, and I broke that fast deliberately, then after that I made up that fast by fasting one day. I do not know whether I should make up one day as I did then or fast for two consecutive months. Do I have to offer expiation (kafaarah)? 

He replied: If a person starts to observe an obligatory fast such as a fast to make up for a day that he missed in Ramadan, or kafaarat yameen (expiation for breaking an oath), or expiation for shaving during Hajj if he shaved whilst in ihraam before exiting therefrom, or other kinds of obligatory fasts, then it is not permissible for him to break that fast unless he has a legitimate shar’i excuse. The same applies to anyone who starts an obligatory action, he is obliged to complete it and it is not permissible for him to interrupt it without a legitimate shar’i excuse. This woman who started to make up a missed fast then broke her fast with no excuse, then made up that day, does not have to do anything further, because she made up one day for another. But she has to repent and seek Allah’s forgiveness for what she did by breaking an obligatory fast with no excuse. 

Majmoo’ al-Fatawa, 20/61.