Does the amount of current drawn by an LED bulb change depending on the brightness? For example, would a bulb at say 25% brightness draw less current and use less electricity than the same bulb at 100% brightness?
Short answer is yes. Longer answer depends on the bulb and how it dims.
Less sophisticated bulbs will just chop the AC power. More sophisticated ones will have a driver that smooths the current to the LED. Either way, the total current will be reduced as it dims.
I have a few Inovelli dimmers with power monitoring and can confirm that the Power consumption goes up when the bulbs are dimmed up, and down when dimmed down. So either the voltage or amperage goes up/down with the dimming functionality.