An heir is someone who receives property, money, or a title from another when the latter person dies. This is not the correct spelling for the phrase putting on airs; however, someone could put on airs about being an heir.
Airs, listed under air in most dictionaries, is a fake way of acting. When plural, one can put on airs or behave in a way that isn’t true. It is used when people act as if they are from a higher class in society, either by making others believe they have more money or are from a recognized family.
Another phrase that uses the same meaning of air is to have an air. Something can have an air of a certain quality, such as elegance or expense.
Examples
In a press release, he has stated that for appointment under compassionate grounds, the age of the legal heir (son or daughter) should be 18 to 35 years. [The Hindu]
He did not put on airs or strut around even once he’d made a name for himself, she said. [The Seattle Times]
When Frisbee golf started putting on airs and became Ultimate Frisbee, it left golf without a lawn-party counterpart. [The Wall Street Journal]
Their items need to carry an air of exclusivity and decadence that the average shopper wouldn’t be able to afford. [Business Insider India]
“He was a quiet, soft-spoken guy, but when he walked in a room he had an air about him where everyone listened.” [Jewish Business News]