It should be "Apple Permanently Closes Charlotte, North Carolina, Store After Multiple Shooting Incidents"
"[Place] store" is a phrase, where [place] serves as an adjective, and "store" is obviously a noun. ("Our Columbia retreat," "Bill's California home," "the Ontario location.")
Grammatically, this structure is the same as that of any "adjective-noun" phrase ("yellow hydrant," "cool dog," "immense shame").
There's no situation in which it would ever be appropriate to comma-separate the adjective and the noun ("I checked out that yellow, hydrant", "what a cool, dog", "it was truly an immense, shame").
If [place] is comma-separated for the purpose of indicating that it's one place within another ("Charlotte, North Carolina"), that use of the comma is denotational rather than grammatical and shouldn't invoke a separate grammatical comma afterward. Typing "Apple Permanently Closes Clarlotte, North Carolina, Store…" would be like typing "we'll vacation at our Columbia, retreat," "I met her at Bill's California, home," or "you should invest in the Ontario, location"). I've never seen any proper headline do this with any location, even with a comma-separated place-within-another-place.
For those that actually care...
Unless a place name is at the end of a sentence and followed by sentence-ending punctuation, whenever you list a city and a state or a city and a country, place commas around the state or the country. The rule applies even when the country or state name is abbreviated. These principles are...
style.mla.org
This column's examples are correct, but those are all situations in which you would finish the sentence or use a comma following the place name anyway, even if the place-within-a-place were replaced with just the name of a place. The column doesn't include any examples involving that adjective-noun-type phrase, or appear to address that concept.