Learn Hardcore German: Kannst du die Heizung anlassen?

Kannst is the 2nd person singular of the modal verb können (to be able to/can). In yes/no questions, German puts the finite verb first, then the subject: Kannst du …? The main (lexical) verb appears at the end in the infinitive: … anlassen?

What exactly does anlassen mean here? Is it “turn on” or “leave on”?

Here anlassen means “to leave something on (keep it running/switched on).” So Kannst du die Heizung anlassen? is asking someone to keep the heating on, not to switch it on.

To switch on: die Heizung anmachen/einschalten
To switch off: die Heizung ausmachen/ausschalten
To turn it up/down (radiator knob): die Heizung aufdrehen/zudrehen, höher/tiefer stellen

Why is it die Heizung? Which case is that?

Heizung is a feminine noun, so its nominative singular article is die. Here it’s the direct object (accusative), and feminine nouns keep die in the accusative: die Heizung.
Examples: das Licht anlassen (neuter), den Fernseher anlassen (masculine).

What does Heizung refer to—heater, radiator, or the whole system?