The church, a powerful voice in the heavily Catholic nation, has called on the electoral commission to publish the true results in "respect of truth and justice".
The church, which deployed some 40,000 electoral observers in all polling centres, cannot say publicly who the clear winner appears to be, as Congo's regulations forbid anyone but the electoral commission to announce results.
The commission accused the church of violating the regulations.
Observers have reported multiple irregularities as this vast, mineral-rich Central African country voted for a successor to departing President Joseph Kabila.
This could be Congo's first peaceful, democratic transfer of power since independence from Belgium in 1960.
The ruling party loyalist whom Mr Kabila put forward as his preferred successor, Emmanuel Ramazani Shadary, already has said he expected to win, while polling before the election had top opposition candidate Martin Fayulu ahead.
The electoral commission's statements on Friday were in a letter to the church and were confirmed by the commission's president, Corneille Nangaa.
In a separate statement, DR Congo's ruling party called the church's attitude "irresponsible and anarchist".
International pressure is growing on Congo to restore internet service, blocked in an apparent attempt to calm election speculation, and release accurate election results, with the United States warning that those who undermine the democratic process could face US sanctions.
"This being a very sensitive, a very tense period, we are concerned that these efforts to silence dissent could backfire considerably when the results are announced," a spokeswoman for the UN human rights office, Ravina Shamdasani, said Friday, also noting intimidation of journalists, observers and rights activists.
"We are watching carefully, and we are calling on all sides to refrain from the use of violence."
Some Congolese have expressed doubt that the first election results will be released on Sunday as expected, as observers have said the internet and text messaging outage has slowed the transmission of election information.
As of Thursday, the electoral commission's president said it had collected results from about 20 per cent of polling stations.
No Western election observers were invited to watch the vote, which was meant to occur in late 2016, after Congo's government was annoyed at international pressure amid concerns that Mr Kabila was trying to stay in power.
Ruling party candidate Mr Shadary, a former interior minister, is under European Union sanctions for a crackdown on Congolese who protested the delayed election.
The UN Security Council was meeting behind closed doors on Congo on Friday at France's request.
Vote-counting continued slowly by hand at more than 175 compilation centres around the country.
At one centre in the capital, Kinshasa, dozens of large bags full of opaque envelopes were piled up while electoral commission agents sat in a courtyard and opened them one by one.
A separate team counted ballots as observers and political parties' witnesses watched.
"We want the electoral commission to be able to work in good conditions because we want peace and we want the true results to be published," said Fiskas Kalombo, a witness for Mr Fayulu's opposition coalition.
"Only the electoral commission can do that and we are here to check that this is what is done."
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