A typical indicator LED will have a spec that represents a typical point along the operating curve. That would look something like “3.3V @ 20mA typical.” Driving this LED above that point will shorten the useful life. You may also get a maximum rating for either current or forward voltage. Exceeding those ratings will dramatically shorten the useful life, generally ending it suddenly in the process. But driving an LED “hot” will make it burn more brightly for a shorter time. The heat dissipated by the junction has to be conducted through the leads, which aren’t very big and aren’t designed to heatsink the package. When the junction runs hot, the light output will also degrade much more rapidly. You may find that your LED won’t light up at all without being overdriven somewhat. If the package of an indicator LED (the 5mm or 3mm types) feels hot, you are definitely overdriving the LED. Illumination-grade LEDs are designed with heat sinks in the package, and it’s normal for these to run hot. The 1W dissipated by the Luxeon Star will warm up a heat sink quickly at room temperature.
Then there’s thermal runaway. The current through the junction will tend to increase as the temperature rises; so current-regulated drivers are preferable to voltage-regulated drivers. But proper thermal management is important too.