Can he observe a fast on behalf of one who is sick during the night in Ramadaan?

Firstly: 

If a person
is sick and unable to fast, one of the following two scenarios must apply:
either his sickness is temporary, so he is allowed not to fast and he has to
make up the fast after he recovers and becomes able to fast, or his sickness
is chronic, in which case he is allowed not to fast but he must feed one
poor person for each day. 

See the
answer to question no. 37761. 

Secondly: 

Fasting can
only be done during the day, from dawn until sunset. The night is not the
time for fasting. Allaah says (interpretation of the meaning): 

“It is
made lawful for you to have sexual relations with your wives on the night of
As‑Sawm (the fasts). They are Libaas [i.e. body-cover, or screen, or Sakan
(i.e. you enjoy the pleasure of living
with them —)] for you and you are the same for them. Allaah knows
that you used to deceive yourselves, so He turned to you (accepted your
repentance) and forgave you. So now have sexual relations with them and seek
that which Allaah has ordained for you (offspring), and eat and drink until
the white thread (light) of dawn appears to you distinct from the black
thread (darkness of night), then complete your Sawm (fast) till the
nightfall”

[al-Baqarah 2:187]

This verse
describes the time for fasting – which is the day – and the time for
breaking the fast – which is the night. It is not valid, under any
circumstances, to make the nights of Ramadaan the time for fasting. 

It is proven
that the Prophet (peace and blessings of Allaah be upon him) forbade
continuous fasting. This was narrated by al-Bukhaari (1062) and Muslim
(1102). 

Continuous
fasting means not breaking the fast at night and continuing to fast night
and day. 

Imam
al-Bukhaari (may Allaah have mercy on him) said: “Chapter on continuous
fasting and those who say that there is no fasting at night because of the
words of Allaah, ‘then complete your Sawm (fast) till the nightfall’
and the Prophet’s prohibition of that as a mercy to them and to avoid
hardship for them. 

Thirdly: 

The basic
principle with regard to physical acts of worship is that the Muslim should
do them on behalf of himself, and they cannot be done on behalf of others.
It is not permissible for anyone to pray on behalf of anyone else, or to
fast on behalf of anyone else, according to scholarly consensus. Rather Hajj
and ‘Umrah may be performed on behalf of one who is unable to do them during
his lifetime, as is stated in the saheeh texts. 

Ibn ‘Abd
al-Barr (may Allaah have mercy on him) said:  

With regard
to prayer, the scholarly consensus is that no one can pray on behalf of
anyone else, whether it is an obligatory prayer or a Sunnah or naafil
prayer, on behalf of one who is living or one who is dead. Similarly,
fasting on behalf of one who is alive is not valid and cannot be done on
behalf of anyone else. There is consensus on all of this and there is no
difference of opinion. 

But if a
person has died and still owes some fasts, this is a matter concerning which
the scholars, both ancient and modern, have differed. End quote. 

Al-Istidhkaar, 3/340 

Conclusion: 

Fasting must
be done during the day, not at night. Fasting at night is not valid. 

It is not
valid for anyone to fast on behalf of a person who is sick. If there is the
hope that the sick person will recover, then he has to make up the fasts
after he gets better. If there is no hope of recovery, then he has to feed
one poor person for each day. 

And Allaah
knows best.