As a general rule, you should assume that your home insurance will not cover a new HVAC unit.
Your HVAC will eventually need to be replaced, owing to general wear and tear. No matter how well cared-for your HVAC unit may be, no machine can be expected to last forever. Even if you only use your HVAC when you absolutely need it — and even if you keep it regularly serviced and maintained —it's eventually going to be more expensive to repair than it is to replace. Because every HVAC has a built-in expiration date, you can't expect your home insurer to protect the purchase.
Likewise, you shouldn't expect your insurer to cover the losses if your HVAC is damaged because you dropped it during installation or encountered similar issues.
You may be able to buy a new HVAC unit from an insurance payout if your home is damaged in a fire, or if the unit is damaged through some other covered hazard. But you can't expect your insurer to simply be your new-HVAC account any time you need a new one.
A homeowner's warranty, on the other hand, may cover a new HVAC purchase or repairs to your current HVAC, depending on your terms. You'll typically cover a service fee, generally in the area of $60. So, don't waste your warranty calls on swapping out air filters or anything that would be cheaper to fix on your own.
Likewise, even in instances where your HVAC may be covered on your home insurance, you won't want to file a claim if it's cheaper to replace the unit on your own than it is to pay the deductible, or if you'd rather not deal with the raised rates. Home insurance is best kept in reserve for costs that you cannot cover on your own. For the average homeowner, HVAC units do not fall into that category.