I think we can probably rule out the story being warning against flight, being that no one was building aircraft at the time, or for quite some time thereafter.

Icarus's fall is a warning about youthful carelessness, and shows the terrible consequences that can bring.

Diadorus Siculus tells two versions of the story, both of which emphasize the recklessness of Icarus causing his downfall:

But when Icarus was disembarking onto the island in a reckless manner, he fell into the sea and perished
...
As for Icarus, because of the ignorance of youth he made his flight to far aloft and fell into the sea when the wax which held the wings together was melted by the sun

Similarly, Pausanias speaks of his clumsy navigation:

The ship of Icarus is said to have overturned, as he was a clumsy helmsman. The drowned man was carried ashore by the current to the island

And Apollodorus refers to him as infatuated, and draws attention to his disregard for his father's instructions:

But the infatuated Icarus, disregarding his father's injunctions, soared ever higher, till, the glue melting, he fell into the sea called after him Icarian, and perished.
Apollodorus Library E.1.13

It would appear that, even 2000 years ago, teenagers thought they were invincible.