Should he pay zakaat al-fitr on behalf of an orphan on whom he is spending?

Firstly: 

If this orphan has wealth, such as inheritance or charity
that is given to him, then his zakaat al-fitr should be paid from his
wealth. Al-Nawawi (may Allah have mercy on him) said: … As for the orphan
who has wealth, his zakaat al-fitr should be paid from his wealth, in our
opinion. This is the view of the majority, including Maalik, al-Awzaa‘i, Abu
Haneefah, Abu Yoosuf and Ibn al-Mundhir.

End quote from al-Majmoo‘, 6/109 

But if he does not have any wealth, rather it is you who are
spending on him and providing what he needs, then you do not have to pay
zakaat al-fitr on his behalf, because zakaat al-fitr is only obligatory upon
the one who is obliged to spend on him. As for the one who is not obliged to
spend on him but is doing so voluntarily, he is not obliged to pay zakaat
al-fitr on his behalf. 

Al-Nawawi (may Allah have mercy on him) said:

If a person voluntarily spends on the maintenance of someone
who is not related to him, he is not obliged to pay his zakaat al-fitr, and
there is no difference of opinion concerning that among us. This is the view
of Maalik, Abu Haneefah and Dawood. Ahmad said: He does have to pay it. …

End quote from al-Majmoo‘, 6/100 

Ibn Qudaamah said: This is the view of the majority of our
companions, because the Prophet (blessings and peace of Allah be upon him)
said: “Pay zakaat al-fitr on behalf of those whom you maintain.”

Abu’l-Khattaab favoured the view that he does not have to pay
his zakaat al-fitr, because he is not obliged to maintain him. This is the
view of the majority of scholars and it is the correct view, in sha Allah.

End quote from al-Mughni, 2/362 

The hadeeth, “Pay zakaat al-fitr on behalf of those whom you
maintain”, was classed as da ‘eef (weak) by many scholars. Al-Nawawi said:
Its isnaad is da‘eef. Al-Bayhaqi said: Its isnaad is not strong. Al-Bayhaqi
also narrated it from Ja‘far ibn Muhammad, from his father, from the Prophet
(blessings and peace of Allah be upon him), but this is also mursal. To sum
up, this phrase, “those whom you maintain” is not proven.

End quote from al-Majmoo‘, 6/68. See also the answer
to question no. 99585. 

Even if we assume that it is saheeh, the spending referred to
in the hadeeth is that which is obligatory, not that which is voluntary. 

And Allah knows best.