Thats going to depend on how the power supply is built, and the quality. but it really goes back to the LED. LED's are not a fixed resistance like normal loads.
Look at this graph (the one on the left) for the LM561C chips. For a given voltage,they draw a specific current, but the current does NOT change based on ohms law. At 2.55 volts or less per LED, you have zero amps drawn, and it peaks out at 3.05 volts at 2 amps.
What that graph doesnt show is what happens to the current as you go past 3 volts per led. The change is non-linear.
In short, very small changes in the voltage result in huge changes in the current flowing. So if the voltage is allowed to fluctuate at all - as will always happen in any PS that isnt CCCV - the current will fluctuate wildly, and if the voltage OR current goes beyond the max, the LED can go into thermal runaway. At a minimum, you reduce its life.
An LED driver needs a very stable output voltage AND current. Even very small fluctuations are bad. That means a quality CCCV type supply that is designed to drive LEDs.