Newry Magistrates Court heard a threat was made from the same account to Khan.
The investigating officer said further posts of racist, xenophobic, sectarian and homophobic nature were also found to have been sent from the same IP address.
During police interviews, Burns confessed to posting the threats, the investigating officer said.
Burns also told detectives during interviews that his political view was far right and that he was against minority groups, including immigrants and members of the LGBT community.
"He agreed with the sentiments contained in the posts. He has antipathy towards minority communities."
District Judge Mark McGarrity refused an application for reporting restrictions in the case.
"Mr Burns, the police will be watching you," Judge McGarrity said.
The defendant was released on £500 (NZ$965) bail with a £1000 (NZ$1931) cash surety on the condition that he had no internet connection and adhered to a 7pm-7am curfew.
The case was adjourned to May 8.