Mostly, no, it won't kill the battery.
Because overcharging a lithium-ion battery destroys it and often results in fire, there is an absolute requirement that the charging stop. So, your battery isn't going to be "cooked" by leaving it plugged in.
On the other hand, lithium-ion batteries don't last as long if they are maintained at 100% charge indefinitely. This is also true if the laptop causes them to be warm all the time. This is why some will avoid leaving the device plugged in continuously.
As far as leaving the device plugged in is concerned, the power supply will be running, as well as the charging circuits (although they should ultimately be doing nothing). Whatever they are doing will generate heat. If they are well designed, they will draw very little power, and generate very little heat. This is the sort of thing the Energy Star rating is about, that is, wasting very little power on standby.
A fair test, then, is to feel the power supply (if it's external) and the device, after they've been sitting a few hours, with the device turned off. If they feel cold, they're consuming almost nothing. To the extent that they feel warm, they are wasting that amount of power. Even in the latter case, they are unlikely to be damaged unless they were poorly designed. And if the device is off, everything else (besides the charging circuitry) is oblivious to the applied power.