Tveit has worked as a journalist for the Norwegian Broadcasting Corporation since 1973. To begin with he covered petroleum development in the North Sea. From 1977 Tveit has been attached to NRK's foreign editorial office ("utenriksredaksjonen").

Cairo 1978

Tveit's first assignment as a foreign correspondent was in Cairo in 1978, a short-term replacement for Kjell Gjøstein Resi.

Beirut 1979–1983

When Tveit started on his first permanent period as a foreign correspondent for NRK, in 1979, he relocated NRK's office from Cairo to Beirut. Tveit stayed in Beirut until 1983.

Amman 1990–1994

During his next period as a Middle East correspondent, 1990–1994, the NRK office was based in Amman.

Amman 2003–2007

Tveit was again Middle East correspondent for NRK from 2003 to 2007, when the office was based in Amman.In 2007 Sidsel Wold was assigned as NRK's new correspondent after Tveit had finished his period in Amman.

Tveit was responsible for the documentary program Brennpunkt: Sporene etter Sharon ("Focus: Tracks after Sharon"), broadcasted by NRK in 2002. This program received the Gullruten award for best TV documentary in 2002.

Tveit wrote three books on energy politics in the early 1970s. From the mid 1980s the theme for his books has been the Middle East. He issued the book Alt for Israel. Oslo-Jerusalem 1948-1978 ("All for Israel") in 1996, and Krig og diplomati. Oslo-Jerusalem 1978-1996 ("War and Diplomacy") in 2005. He was awarded the Brage Prize for non-fiction in 2005, for Krig og diplomati.