Anthony was a DHARMA Initiative architect. He may have been killed along with other members of the DHARMA Initiative in the Purge, or he may have left the island sometime between 1977 and The Purge.
On the Island
Anthony was seen at the cafeteria having his breakfast one morning. He also present at "the vote" to determine if Sayid should be executed. Later that night a burning DHARMA van crashed into building 15, one of the houses at the Barracks, and Anthony was seen pulling out a fire hose to put out a fire with Juliet and Clive. ("He's Our You")
Since only "Anth" was shown on Anthony's jumpsuit in "He's Our You", there's not enough evidence to support the name Anthony. However, his name and job title were both finally confirmed in a behind-the-scenes photo from the same episode. According to the photo, Anthony was a DHARMA Initiative architect, a job title apparently unique to him.