The Eyrie Pyramid will be the name of the biggest and tallest building in Manhattan in 2198. Technically, it is the super-modernized Eyrie Building. It will still be crowned by Castle Wyvern and the Manhattan Clan will continue to reside there. The building will be owned by the Lexington-Xanatos Corporation.
Any connections with the symbol of the Illuminati remain unclear. 
In Puck's illusion, the Eyrie Pyramid was a huge building complex in Manhattan with the Eyrie Building at its core and culminating in a massive pyramid-like structure placed over Castle Wyvern. It served as a sort of broadcasting screen, allowing the Xanatos Program, masquerading as Xanatos, to deliver public messages to the few remaining citizens of New York. It is unknown if the true Eyrie Pyramid of the future will resemble Puck's vision of the building, though it seems likely. 
- "Future Tense" (First Appearance, Illusion)