Forward voltage versus current: -
Below maybe 1.5 volts you might not see any residual glow from an LED with your eyes and as expected the current (Y axis) is sub 1mA. If you doubled that voltage to 3V, the current would be off-the-scale in the Y axis and, for a normal standard LED this spells the end of the road.
All LEDs have this knee voltage thing and this is why we choose to control the current through LED rather than try and control the voltage across it.
Luminous intensity versus current typical example: -
On this example (right hand graph) the luminous intensity increases linearly with current but as currents rise beyond the scope of that graph you get less payback. Here's an example showing light output versus current and, quantum efficiency versus current: -
Efficiency reaches a peak at some moderate amount of current but the payback gets less as you rise above this value. Always read the data sheet on a specific LED - the graphs above are very generic.