Most LED "bulb replacements" have their driver in them (in theory you could make one "driverless" using 40ish LEDs at 3-ish Vf = 120ish Vf total (to match 120V US supplies). Those drivers are typically only designed to operate either "off" or "120ish" volts, and they boost or buck accordingly for the desired output parameters. Their compents are rated accordingly, if you have designed around a controller chip designed to operate at 10% duty cycle at 120V input, then if you operate them too far out of range (really low for instance), the chip might be running 100% while trying to "boost" the circuit (just a for-instance)... it never gets enough voltage of course to generate the desired output voltage/current, so although the LEDs themselves will be fine (LEDs don't care about "too-low" voltage), the controller IC may burn itself out trying to make it work, making your bulb worthless.
That's just an example possibility, and if it's just induced low voltage/hig-millivolt levels, it's unlikely the IC will be able to run at much of a duty cycle either (you can "boost" from below 1V in some circuits to a few volts)... so basically the "start" circuit boosts the millivolts to 3.0V or so (which typical ICs use nowadays)... which wakes up the IC, it tries to do it's thing, gets a little boost out to the LEDs, and then the whole circuit probably stops/collapses electrically until the next "start" cycle begins, so it'd be a fractional duty cycle for the whole circuit, not going to hurt anything.
The "no-dimming" problem is totally different, many dimmers work by "chopping" the current to the fixture (or they may cut the voltage level, with the same result/problem below), if it's a smooth 60hz sine wave, and you dim 50%, it "chops" out every other cycle for instance. Incandescent bulbs don't care much, they just cool a tiny bit in the off part then back up in the on part, but it's not a major percentage of their operational range.
However, your LED circuit cares a LOT about this (either chopping or dropping voltage)... if it's chopped 50%, then the driver has only 50% of it's operating power to work with to the LED... remember, it wants to drive the LED 100% (unless designed for dimming), so the built-in circuitry has to work twice as hard to boost the output. If the parts are designed around a 120V/1A input, now it has to take 2A during the "on" cycle to make up for the "off" cycle, potentially burning out power controllers, etc. AND/OR the controller IC runs twice as hard as designed, burning itself out, or all the above... bottom line, it's working outside it's design specs and going to die a premature death.
<edit> A driverless (say 140Vf worth of LEDs (or LEDs and resistors) in series (to prevent surge voltage damage) fixture on the other hand would have no issues with dimming, if the dimmer chops the current, the LED runs at a fraction of it's full "on" state, if it drops voltage, the LEDs run below their max Vf, but are fine as well. It's the control circuits that are the issue (although a dimmed control circuit could potentially cause damaging surges as a consequence).
