Can a weak power supply damage connected electronics?
Using a 1 Amp power supply to power an LED strip that required 8 Amps is likely to damage the power supply, unless the supply includes over-current protection.

In any case, the power supply's output voltage will drop when you try to draw excessive current from it.  This will cause the LED strip to not work (or to not work correctly), but should not cause any permanent damage to the LED strip.