I thought it would be a good idea to make a complete list of limited edition Dreamcast consoles. It doesn't appear that anyone has done this before. All the lists I've found online have been incomplete. Here's what I've found so far. I'm being very broad with the term "limited edition" and including official replacement shells, promotional Dreamcasts, and even rental versions. If you see any missing, just let me know and I'll add them.
Biohazard: Code Veronica
Japan-only limited edition.
Biohazard S.T.A.R.S.
Japan-only limited edition.
Christmas Seaman
Japan-only limited edition.
Clear (Replacement Shell)
Official replacement shell sold through the Sega D-Direct website in Japan.
Clear Black (Replacement Shell)
Official replacement shell sold through the Sega D-Direct website in Japan.
Clear Blue (Replacement Shell)
Official replacement shell sold through the Sega D-Direct website in Japan.
Clear Yellow (Replacement Shell)
Official replacement shell sold through the Sega D-Direct website in Japan.
Coca-Cola
From what I can gather, this Dreamcast was given away in a drawing by a Japanese restaurant called "Jonathan's".
CSK Health Insurance
This Dreamcast and Dreameye camera set was loaned to hospital patients so that they could video chat with their doctors. It also included a special version of Dream Passport. More info here.
Divers 2000
The Divers 2000 Dreamcast was a console/TV combo exclusive to Japan. The set included a Dreameye camera. Sega marketed this as an in-home video chat system.
Gold
A special golden Dreamcast given away in a Pro Yakyuu Team De Asobou Net contest. Only around 5 are known to exist.
Hello Kitty (Blue)
Japan-only limited edition.
Hello Kitty (Pink)
Japan-only limited edition.
Infogrames / Game Planet
Unknown origins. Possibly given away in a contest?
K-Office
Unknown origins.
Lawson Ticket Station
Given away in a contest by a convenience store in Japan called Lawson. The cost was 2000 YEN to enter the contest.
Maziora
Dreamcast created as a collaboration with the Maziora paint brand. Only 500 were made and were available for order online or won at some racing events. 200 of these were signed by Tadahiko Tair, a Japanese race car driver.
Metallic Silver
Japan-only limited edition available only from the Sega D-Direct website.
Milka
Unknown origins. Likely won in a contest.
Millennium Soldier: Expendable
Five of these were given away in issue #2 of Dreamcast Monthly magazine in the UK.
Partners
A special Dreamcast sold only to members of the Sega Partners club.
Pearl Blue
Japan-only limited edition available only from the Sega D-Direct website.
Pearl Pink
Japan-only limited edition available only from the Sega D-Direct website.
Pure Black
Japan-only limited edition available only from the Sega D-Direct website.
R7
Japan-only limited edition.
RX-78
Japan-only limited edition available only from the Sega D-Direct website.
Sakura Wars
Japan-only limited edition.
Sakura Wars "Specialty Shop"
This version of the Sakura Wars Dreamcast was made to commemorate the grand opening of the Sakura Taisen Specialty Shop in Shinsaibashi, Japan.
Seaman
Japan-only limited edition.
Sega Sports
The only limited edition Dreamcast released in the US.
Sonic 10th Anniversary
Special edition Dreamcast made to celebrate Sonic's 10th anniversary. Only 50 were made; 40 of which were sent out to SoE (Sega of Europe) employees and 10 were sent to France for magazine giveaways. One of these consoles exists signed by Yuji Naka.
Swatch
This Dreamcast was created for the partnership between Sega and Swatch which sadly never amounted to anything. Likely only one of these Dreamcasts exists, even though there seems to have been plans to make more.
Toyota
A variation of the Dreamcast sold in Toyota dealerships. It included a special version of Dream Passport with links to Toyota websites.
Trial / Rental
This Dreamcast was a short-lived collaboration between Japanese department store Tsutaya and Sega to rent Dreamcasts.