Referring to 'heating' is certainly common, such as a heating/cooling mechanic or a heating system. And I'll grant that there may be regions where the phrase "turn on the heating" is used, but not in my travels that I can recall (and I've been around the US quite a bit). Regardless, I think we can agree that 'turn on the heat (heater)' should be acceptable.
SueWaller: To my ear, 'start the heating' sounds unnatural. It'd be understood, but I doubt it's what a native speaker would say. In another context 'accendere' might well mean 'to start', but it can also mean to 'ignite' and I don't think you'd want to say "We want to ignite the heating". So for me, context is everything and 'start' just sounds incorrect.
If 'accendere' is 'to turn on', what would be the Italian for 'to turn up'? The reasnoning (faulty, as it may turn out to be):
'accendere' is also translated as 'to raise', which contextually is akin to 'turn up' or 'increase', plus
'accendere' is similar to 'ascend', which itself is closer to 'move up' than to 'turn on'.
PS: The comments by Nonna and Sedona, showed on this page only after i posted mine. Apologies for the repetition.
PPS: just reached 'accendere la macchina' - I surrender.
"Accendere" actually comes from the word "to light" or "to ignite" (and means these things in Italian), no relation to "ascend." I can't think of any contexts where it would mean "raise." It has similar roots to "candle" and "incandescent" in English: https://en.wiktionary.org/wiki/accendo#Latin
To turn up the heat, you would use the verb "alzare." Note that this would be "turning up the heat" in a literal sense, as the English expression has several figurative meanings.