Do LEDs Need UVLOs (Undervoltage-Lockouts)?
No.

LEDs are not damaged by low voltages.

If you are damaging LEDs, you must be driving them beyond their rated currents.  Show your circuit to receive advice.

In general, very few electronic components are damaged by undervoltage.  Some microprocessors can mis-execute in a brownout condition, which could have undesirable effects depending on the application.  And Li-Ion batteries should not be discharged to too low a voltage.