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 Ramadaan 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.
38752 and
39991. 

If a person breaks a fast that he is observing to make up for
a missed Ramadaan 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 Allaah 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 Ramadaan 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 Ramadaan fast only has to fast
one day, because the obligation to make up the replacement day is not
something that was enjoined by Allaah. It was narrated in a saheeh report
that the Prophet (peace and blessings of Allaah be upon him) made up
that day of Ramadaan so it is not permissible to add anything else to that
without any textual evidence or scholarly consensus. 

The Shaykh (may Allaah 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 Ramadaan, 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 Allaah’s
forgiveness for what she did by breaking an obligatory fast with no excuse. 

Majmoo’ al-Fataawa, 20/61.