From Season 4, House has three new doctors as part of his team and the previous three just appear off an on. Why didn't the show's creators update the intro to show the new team? It always shows Cameron, Chase and Foreman.
Season 4 takes place after House fires his entire team and must start playing his game to find a new one.
During this season however, the production crew knew that the old team members would still be a part of the show in some capacity. Foreman eventually rejoined the team, Chase became the head of Surgery, and Cameron became the head of the ER.
With that spoiler out of the way, the crew themselves did not know who was going to be voted off House's Game of Jobs. This combined with the Writer's Strike at the time made the series as chaotic as possible. This is what adds to the fact that the opening sequence did not change to reflect the new cast members, merely putting them in the credits at the start of the actual episode.
In fact, if memory serves this continues until at least season 7 where the intro completely changed to reflect Jennifer Morrison's departure from the series and features brand new sequences in the intro including spots for Peter Jacobson and Olivia Wilde in the intro who were upgraded to full cast members.