Can I claim form E-589CI on my tax return? I searched previous answers but they were dated 2019.

For North Carolina?

Why do you want to include it?  Did you not pay sales tax on the purchase?  

A new HVAC system would be a capital improvement to your home.

Form E-589CI, Affidavit of Capital Improvement, may be used to substantiate that a contract, or a portion of work to be performed to fulfill a contract, is to be taxed for sales and use tax purposes, as a real property contract for a capital improvement to real property.

The certificate is proof that you paid sales tax on your new HVAC. Now the story can go a couple of paths from here:

1. You are itemizing deductions and want to include the sales tax to help determine your state and local taxes paid.

2. You want a credit for installing the HVAC

• A new HVAC might qualify for a federal energy credit on form 5695 if it is meets the rules. See Energy Tax Credit: Which Home Improvements Qualify
• If this HVAC is on a rental, it would be depreciated as an asset.