Did anyone ever claim responsibility for downing the plane?
Yes. Link to Swordmakers post. Post 34
"In the absence of explanations, theories abounded. One focused on a fax sent Wednesday to an Arabic language newspaper in Beirut warning of an attack. State Department and CIA officials confirmed they had received copies of the fax Thursday. The message said "tomorrow morning we will strike the Americans in a way they do not expect and it will be very surprising to them," according to one official. A counterterrorism source familiar with the fax said that it was sent at 11 a.m. New York time Wednesday, more than nine hours before the bombing. But a CIA source said that the agency "does not attach too much significance" to the fax.
"The fax, written in Arabic, ends with the following threat: "The Mujahadeen will respond harshly to the threats of the stupid American president. All will be shocked by the magnitude of the response. The determining of the place and time are in the hands of the Mujahadeen. The invaders must get ready to leave alive or dead; and their rendezvous will be morning, and isn't morning near." U.S. News and World Report magazine, in the July 29, 1996 issue, identified the group who sent the fax as, "The Movement of Islamic Jihad/The Jihad Wing of the Arabian Peninsula." [Editors Note: 8pm Eastern Time in New York is early morning in the Persian Gulf.]"
Newsday.com, July 19, 1996
. . . Senior Iranian sources close to the fundamentalist regime in Tehran claimed this weekend that TWA flight 800 was shot down last month by one of three shoulder-fired Stingers of the type used by Islamic guerrillas during the Afghanistan war. The sources said the missiles arrived in America seven months ago after being shipped from Karachi via Rotterdam and on to the Canadian port of Halifax. They claimed an Egyptian fundamentalist group backed by Iran was responsible for smuggling the weapons across the Canadian border into the United States. The group, the Gama'a al-Islamiya, comprises followers of Sheik Omar Abdel-Rahman, a blind Egyptian cleric jailed in the United States over the 1993 New York World Trade Center bombing."The Times of London, August 27, 1996
" . . . at least one terrorist has claimed credit for the TWA 800 bombing. World Trade Center bomber Ramzi Ahmed Yousef told authorities his group is responsible. Yousef's claim has not been made public, but it is in the FBI file."The American Spectator, Sept. 1997
If you can read Arabic, here is the Al Hayat article threatening the attack.