Drinking from Zamzam is mustahabb, not obligatory

Firstly: 

Drinking Zamzam water is not obligatory, rather it is
mustahabb, but it is not mustahabb only after praying the two rak’ahs behind
the Maqaam, rather drinking from Zamzam is mustahabb at all times. 

Shaykh al-Islam said in Majmoo’ al-Fataawa (16/144): 

It is mustahabb to drink
Zamzam water and to drink one’s fill of it, and to make whatever Islamically
prescribed du’aa’s one wants when drinking it. 

Al-Muwaffaq said in al-Mughni: 

It is mustahabb to come
to Zamzam and drink from its water for the one who wants to, and to drink
one’s fill of it. Jaabir said, describing the Hajj of the Prophet
(peace and blessings of Allaah be upon him): “Then he came to Banu ‘Abd
al-Muttalib, who were providing water, and they gave him a bucket, and he
drank from it.” 

Al-Nawawi said in al-Majmoo’: 

Al-Shaafa’i, his
companions and others said: it is mustahabb to drink Zamzam water, and to
drink a great deal of it, and to drink one’s fill. It is mustahabb to drink
it for whatever benefits one wants in this world and the Hereafter. If a
person wants to drink it for forgiveness or healing from sickness etc, he
should face the qiblah and mention the name of Allaah, then say, “O Allaah,
I heard that Your Messenger (peace and blessings of Allaah be upon him)
said: ‘The water of Zamzam is for whatever it is drunk for.’ O Allaah, I am
drinking it so that You will forgive me so forgive me, O Allaah,” or: “I am
drinking it to seek healing from sickness, so heal me, O Allaah.” And so on.
It is mustahabb to drink in three draughts as in the case of any other
drink. And when he is done he should praise Allaah. 

Shaykh Ibn Baaz said in Majmoo’ al-Fataawa (16/138): 

It is mustahabb for the pilgrim doing Hajj or ‘umrah to drink
from the water of Zamzam if he is able to do so. 

Based on this, if you do ‘umrah when you are fasting, there
is no sin on you if you do not drink from Zamzam, and you can drink from it
after you break your fast. 

Secondly: 

If you are traveling to Makkah for ‘umrah, it is permissible
for a traveler either to break his fast or to fast, according to scholarly
consensus, but they differed as to which is better. We have already
mentioned in the answer to question no.
20165 that what is better is what is
easier. For those for whom fasting when traveling is not too difficult, then
fasting is better, and for those for whom fasting is too difficult whilst
traveling, not fasting is better, especially for one who is doing ‘umrah,
because he needs physical strength in order to do ‘umrah properly, with
du’aa’ and proper focus and humility. 

Some people who do ‘umrah make it difficult for themselves by
doing ‘umrah whilst fasting, even though fasting is very difficult for them,
which affects the way they do ‘umrah. It was proven that the Prophet
(peace and blessings of Allaah be upon him) used to break his fast on the
Day of ‘Arafah. 

Al-Shawkaani said: 

Fasting the day of ‘Arafah is mustahabb for everyone, but it
is makrooh for those who are in ‘Arafah for Hajj. The reason for that is
that it may lead to the pilgrim becoming too weak to offer du’aa’ and
remember Allaah (dhikr) on the Day of ‘Arafaah in that place and to do the
actions of Hajj. 

It was proven in al-Saheehayn that the Prophet
(peace and blessings of Allaah be upon him) forbade singling out Friday for
fasting. 

Al-Nawawi said: 

It is makrooh to single
out Friday for fasting unless that coincides with one’s habit [e.g. fasting
alternate days], but if a person fasts the day after or the day before as
well, or it coincides with his habits, such as if he vowed to fast the day
when he was healed forever, and that happens to be a Friday, then it is not
makrooh. 

The scholars said: The reason why it is forbidden is that
Friday is a day of du’aa’, dhikr and worship, such as doing ghusl, going to
the prayer early and waiting for the prayer, listening to the khutbah and
doing a lot of dhikr afterwards. Allaah says (interpretation of the
meaning): 

“Then when the (Jumu‘ah)
Salaah (prayer) is ended, you may disperse through the land, and seek the
Bounty of Allaah (by working), and remember Allaah much”

[al-Jumu’ah 62:10]

And there are other acts of worship to be done on this day,
so it is mustahabb not to fast on Friday, because that helps one to do these
duties in an energetic manner and to enjoy them without getting tired or
bored. This is similar to the situation of the pilgrim on the Day of ‘Arafah
in ‘Arafah – the Sunnah is for him not to fast as stated above, for the same
reason. If some one were to say: if a person fasts another day as Friday
that will still make him tired and affect his worship. The answer is that by
fasting an extra day he will gain reward that will make up for any
shortcomings in his worship on Friday. This is the reason why it is
prohibited to single out Friday for fasting. 

Shaykh Ibn ‘Uthaymeen was asked in Fataawa Arkaan al-Islam
(p. 464) about a traveler who reaches Makkah fasting: should he break his
fast in order to have more strength to do ‘umrah? 

We say: when the Prophet (peace and blessings of Allaah
be upon him) entered Makkah on the twentieth of Ramadaan in the year of the
Conquest, he broke his fast. 

It is proven in Saheeh al-Bukhaari that the Prophet
(peace and blessings of Allaah be upon him) did not fast for the rest
of the month, because he was traveling. The journey of the pilgrim doing
‘umrah does not end when he reaches Makkah, and he does not have to refrain
from eating etc if he comes not fasting. Some people may continue to fast
even when traveling, because fasting whilst traveling nowadays is not a
hardship for the ummah, so they continue to fast whilst traveling then come
to Makkah feeling very tired, and they wonder whether they should continue
their fast or not, or should they delay ‘umrah until after they have broken
the fast, or break the fast in order to do ‘umrah straight after reaching
Makkah?

We say to them in this case: it is better for you to break
your fast in order to do ‘umrah as soon as you reach Makkah, when you are
still feeling energetic, because the Sunnah is for the one who comes to
Makkah for pilgrimage to hasten straight away to do these rituals. When the
Prophet (peace and blessings of Allaah be upon him) entered Makkah for
the purpose of pilgrimage he would hasten to the mosque, and he used to make
his camel stop at the mosque and go in to do his rituals that he had come to
do. Not fasting so that you will be able to do ‘umrah in an energetic manner
during the day is better than continuing to fast and then doing ‘umrah after
you break your fast at night. It was proven that the Prophet (peace and
blessings of Allaah be upon him) was fasting when he was traveling to
conquer Makkah, then the people came to him and said: “O Messenger of
Allaah, fasting is proving too hard for the people, and they are waiting to
see what you will do.” And that was after ‘Asr. So the Prophet (peace
and blessings of Allaah be upon him) asked for a drink of water and drank it
whilst the people were looking on. So he broke his fast whilst traveling,
and he even broke his fast towards the end of the day, all in order to show
his ummah that that is permissible. Making oneself fast whilst traveling
even though it is too difficult is undoubtedly contrary to the Sunnah, and
the following words of the Prophet (peace and blessings of Allaah be
upon him) apply in this case: “It is not righteousness to fast whilst
traveling.” 

If your fasting whilst traveling will affect your performance of ‘umrah, then it is better for you to break your fast and make that day up later on.