Are you sure the LED is wired correctly? (Cathode to ground, anode to FPGA.)
If you move the DVD player really close to the circuit, does it work?
The FPGA may not be able to drive the IR LED hard enough to communicate. Typically, at least a driver transistor is used between the FPGA or microcontroller and the LED.
There is a trick you can use to estimate the magnitude of the IR output. Many digital cameras and camcorders are sensitive to IR light. Look through the viewfinder and press a button on the handheld remote. Do you see a flash? If so, then do the same with your circuit. No flashes or weak flashes mean you have problems with the IR LED driver stage.