There usually isn't enough power in the KB scan to power LEDs so you'll see them. You'll also likely need resistors to properly match resistance.
Also, if you installed LEDs backwards, you might get light from them, but you'll lose NKRO. If you install them "properly" you'd have NKRO, but the LEDs wouldn't light up.
The diodes are there to prevent current from flowing. If there's current in them, then something has gone wrong.
I don't think there is a danger of burning out the LEDs, There is a very real danger of burning out the controller chip immediately once power is applied.
I could be wrong though, I haven't had any EE classes.