AThe answer to this question is model dependent. LCD display models will normally continue to operate well below their specified minimum input levels, but the display's contrast will gradually diminish, and more importantly, the readout accuracy is no longer guaranteed.
For LED display models, the intensity of the display will diminish, but the readout will remain fairly accurate down to the level at which the display becomes so dim that it becomes unreadable. In applications in which the display is required to be totally off at times, be sure to drive the meter with components which bring the input signal all the way to zero.