Is there a shuttle bus that will take you from Tsim Sha Tsui to Futian Checkpoint?

No direct public transport is avaiable. Take MTR Tsuen Wan Line from Tsim Sha Tusi to Yuen Long Station, Exit E. Then, change to bus line B1 to Ha Wan Fisherman San Tsuen Station, and walk north to Lok Ma Chau Station. You will reach Futian Checkpoint after crossing the border. :)