Jason Taylor's absence last year means he is not the longest-tenured Dolphin, consecutively. Though he has spent the most time as a Dolphin overall.
John Denney is currently in his 7th year with the team (he was picked up in '05).
But there are two players left from the 2004 draft: Yermiah Bell and Vernon Carey. Either could be considered as the right answer, but I'll go Carey because he was drafted before Bell, and started immediately (Bell was injured and played only a little, later in the year)