June 6, 2008 to December 31, 2019
This ENWorld thread declares that D&D Insider was live as of Jun 6, 2008. An archived copy of the D&DI News page, also dated June 6, 2008, announces the launch of the new Dragon and Dungeon. That issue's Ampersand column names that date "D&D Day", and signifies the the official launch of the D&D 4e core rulebooks.
However, as that article states, most of the D&D Insider features were not available at launch. D&D Insider at this point consisted of Dragon and Dungeon magazines. According to this Gleemax forum post by D&D Insider executive producer Ken Troop, dated June 7, 2008, the Compendium and other tools were not available at launch.
The service was meant to launch fully complete as a paid service, but was initially free due to its incomplete status. As late as this Ampersand article dated May 7, 2008, WotC were still hoping to have at least the Compendium ready at launch, with the service charging a price of $14.95/month following an initial period of beta testing and introductory pricing.
The last Dragon/Dungeon issues were December 2013, but the service was still available. By July 29, 2014, for example, you could still subscribe to D&D Insider, at $9.95/month or $5.95/month with annual subscription. It consisted mainly the Compendium, character builder, monster builder, and the Dragon and Dungeon magazines. You could still subscribe to D&D Insider after the release of D&D 5th edition.
According to this article, D&D Insider was scheduled to finally shut down on December 31, 2019. This Reddit thread shares an email notice that the service would be free of charge during December 2019 and anyone who already paid for that month would be offered refunds. You can see in the web archive that the Compendium search page was still online as of December 22, 2019, but unavailable by May 24, 2020.
The reason given for the shutdown was that Microsoft was ending support for the Silverlight platform, which the D&D Character Builder relied on after moving from a Windows-only desktop program to a web-based Silverlight version. The advantage of the Silverlight version was that it opened up Mac support, but a major reason for the change was probably to prevent people from simply paying for a single month and downloading all the character data, or just pirating a copy.
Several major features were ultimately never completed, including a character visualizer, dungeon builder, 3D virtual gametable, and 2D virtual gametable.