Where does the money for the Teen Titans come from. Who built and pays for the tower? I know Robin on the team has access to a lot of money.
There are too many versions of the Teen Titans to easily answer this question. In most continuity versions of the team (Pre-Crisis, for example) can be answered by saying Bruce Wayne (Batman) provided the funding for most of the land acquisitions and technical resources (assuming they were not built by the Titans themselves) for the team in its early days. Some continuity versions of the team were also directly supported by the same agencies which supported the Justice League of America.
The First Teen Titans
Robin (Dick Grayson), Kid Flash (Wally West), and Aqualad (Garth) — the sidekicks of Justice League members Batman, the Flash, and Aquaman — teamed up to defeat a weather-controlling villain known as Mister Twister in The Brave and the Bold #54 (July 1964) by writer Bob Haney and artist Bruno Premiani.
They subsequently appeared under the name "Teen Titans" in The Brave and the Bold #60 (July 1965), joined by Wonder Woman's younger sister Wonder Girl (Donna Troy) in her first appearance. After next being featured in Showcase #59 (December 1965), the Teen Titans were spun off into their own series with Teen Titans #1 (cover-dated February 1966) by Haney and artist Nick Cardy.
The original Teen Titans were financed and supported by Bruce Wayne (Batman) and he gave them their first headquarters called the Titan's Lair.
Titans Lair was once a records storage facility owned by Wayne Enterprises. Bruce Wayne allowed his sidekick Robin the use of the empty silo during the time when Robin helped to form the original Teen Titans.
The one-level cavern is located beneath a cliff adjacent to the Gotham City River. Special emergency exits exist at the base of the cavern, which allowed the Titans the ability to come and go in secret. It also enabled Aqualad quick access to the nearby river and Atlantic Ocean. The Lair contained a concealed helipad for the Titans Helicopter as well as living quarters, cargo lift elevators, laboratories, a warehouse and a mechanic's shop.
This headquarters was later abandoned after being compromised by the villain Doctor Light.
The city may sponsor them in return for their service/role as guardians and protectors. Robin may also be referred to as Batman's sidekick, so they may be partially sponsored by Batman himself. Depends on which version of the Teen Titans you're talking about, as Monty129 has, rather, importantly pointed out.