What are the cases in which the funeral prayer cannot to be offered for the deceased?

The funeral prayer is a communal obligation (fard kifaayah)
that must be offered for anyone who dies and was apparently Muslim, even if
he or she committed major sins. 

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

Offering the funeral prayer for the deceased is a communal
obligation with no difference of opinion among us, which means there is
(scholarly) consensus.

End quote. al-Majmoo‘, 5/167 

There is no shar‘i evidence to suggest that there was any
exception to offering the funeral prayer for any Muslim except the shaheed
(martyr) who was killed in battle. 

Ibn al-Qayyim (may Allah have mercy on him) said: 

The funeral prayer is not to be offered for the martyr who
was killed in battle, because the Messenger of Allah (blessings and peace of
Allah be upon him) did not offer the funeral prayer for the martyrs of Uhud,
and it is not known that he offered the funeral prayer for any of those who
were martyred fighting alongside him in his campaigns. The same is also true
of his successors, the Rightly Guided Caliphs, and their governors and
commanders.

End quote from Zaad al-Ma‘aad, 3/217 

See also the answer to question no.
14012 

With regard to the miscarried foetus, if it had not reached
four months’ gestation, no funeral prayer is to be offered for it because it
is not regarded as human and the soul had not yet been breathed into it. If
it is miscarried after four months of pregnancy, then the funeral prayer is
to be offered for it. 

The scholars of the Standing Committee for Issuing Fatwas
said: 

If the foetus had not reached four months’ gestation, then it
is not to be washed and the funeral prayer is not to be offered for it; it
is not to be given a name and no ‘aqeeqah is to be offered for it, because
the soul had not yet been breathed into it. 

End quote from Fataawa al-Lajnah ad-Daa’imah, 8/408 

There are some individuals whose situation is not clear to
others, so people think that they are Muslims and offer the funeral prayer
for them, when in fact they are not Muslims, such as apostates and
hypocrites. 

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said: 

The apostate of any type is not to be treated in the same
manner as one who is originally a kaafir; rather he has to return to Islam.
If he becomes Muslim again, all well and good; if he does not become Muslim
then he is to be executed as a kaafir. He is not to be buried with the
Muslims and the funeral prayer is not to be offered for him. 

End quote from Fataawa Noor ‘ala ad-Darb, 14/6 

With regard to the hypocrite, who is the one who conceals
kufr whilst outwardly appearing to be Muslim, if it is known that a person
is a hypocrite, then the funeral prayer is not to be offered for him. 

Allah, may He be exalted, says (interpretation of the
meaning):

“And never (O Muhammad SAW) pray (funeral prayer) for any
of them (hypocrites) who dies, nor stand at his grave. Certainly they
disbelieved in Allah and His Messenger, and died while they were Fasiqoon
(rebellious, – disobedient to Allah and His Messenger SAW)”

[at-Tawbah 9:84]. 

Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him)
said:

If it is known that a person is a hypocrite, it is not
permissible to offer the funeral prayer for him or pray for forgiveness for
him. If no such thing is known about a person, then the funeral prayer
should be offered for him. If an individual knows that a person was a
hypocrite he should not offer the funeral prayer for him, but those who did
not know that he was a hypocrite may offer the funeral prayer for him. 

‘Umar (may Allah be pleased with him) used not to offer the
funeral prayer for one for whom Hudhayfah did not offer it, because during
the campaign to Tabook, he knew who the hypocrites were who planned to kill
the Messenger of Allah (blessings and peace of Allah be upon him). End quote
from Minhaaj as-Sunnah, 5/160 

And Allah knows best.