Thai spokesman Sek Wannamethee said Thailand had asked New Zealand officials to clarify Mr Ekaphop's refugee status.
"Mr Ekaphop is exploiting his status granted by the New Zealand Government to conduct political activities which have reverse impact on Thailand's security," he said.
"Such a movement is considered an obstacle to the peace-building process and the good relationship between both countries. Therefore, Thailand requested New Zealand to revoke his status in order to stop his actions against the law."
The Ministry of Foreign Affairs confirmed officials had met with Thai counterparts to "explain New Zealand's refugee resettlement policy".
People who have been granted refugee status by the United Nations refugee agency could resettle in New Zealand under its quota refugee system.
The Bangkok embassy and immigration officials refused to comment on the matter.
Immigration NZ said it could not discuss the case "on privacy and legal grounds".
The charges against Mr Ekaphop stem from a speech he made to a rally in 2013 in support of the Government at the time. He fled to Cambodia after May last year, when a military junta overthrew the elected Government.
Criticising Thailand's monarchy was punishable by up to 15 years' jail. Thai media said coup leader and Prime Minister Prayuth Chan-ocha had increased prosecution of lese majeste suspects.
Foreign Minister Murray McCully did not respond to questions about whether the matter would affect relations with Thailand, which is our 10th-largest trading partner.
Q & A
What did Ekaphop Luera do?
The Thai national was charged with insulting his country's monarchy at a political rally in late 2013, which is against the law. Since a military coup in May 2013, the new military leadership has increasingly pursued dissidents.
Why is he in New Zealand?
He was granted refugee status by the UNHCR last year and fled here via Cambodia to escape persecution in his home country.
What are lese majeste laws?
The laws forbid anyone from threatening or insulting the Thai monarchy and carry a maximum 15-year jail sentence.