Make sure you have the latest updates to FRC Java or C in Eclipse, and make sure your roboRIO has the latest firmware.
Once code has loaded onto the robot, you do not need Eclipse running or connected to the driver station for code to work. The code is executed locally on the roboRIO; the computer is only a communication device at that point (unless you’re offloading something like camera code onto it, etc).
If resetting the roboRIO doesn’t work (can be done from the DS app, there’s a button for it), then try reloading code using a tether. If that doesn’t work, you may need to try reimaging your roboRIO.
Sometimes right after loading code, the robot code just doesn’t turn green for me, and I have to reboot the roboRIO / power cycle the robot to make it work. It usually goes away pretty quickly though.