Arunachalam Chrishanthakumar, also known as AC Shanthan, 50, and Goldan Lambert, 29, appeared at Westminster Magistrates Court, central London, charged under the Terrorism Act 2000.
The pair, who both gave addresses in south London, are accused of providing support to a "proscribed organisation".
Chrishanthakumar was remanded in custody until a further hearing at the same court on August 9.
Lambert was released on bail but was ordered to appear on the same date to hear if the case will be sent to a higher court.
London's Metropolitan Police said Chrishanthakumar faces five charges, including helping to arrange a meeting he knew was "to support a proscribed organisation, namely the Liberation Tigers of Tamil Eelam" in July last year.
Another charge relates to an alleged address of a meeting in London to encourage support for the Tigers, and allegedly receiving 1,500 pounds (2,230 euros, 3,003 dollars) in January 2005 "intending that it be used or having reasonable cause to suspect