Applying 12v to a 24v heater cartridge won't damage anything, but you may have severe issues reaching and maintaining your target temp. A standard E3D heater cartridge is 40w. When you run a 24v cartridge on 12v, you only get 10w of heater power. Here are some rough estimates on where your hot block heat goes:

• Uninsulated hot block air losses: ~20w depending on airflow


• Insulated hot block air losses: ~5w depending on airflow


• Filament melt power at typical print speeds: 0.3-3w depending on filament and print speed


• Heat conducted up the heat break: 1-3w maybe, hard to quantify

I've never heard of anyone using less than about 16w to print. (Two 8w power resistors.) 20w is the lowest vaguely normal hot end power. I think you'll struggle to print.