Hackers were reported to have infiltrated IT systems - but the gallery says its works are safe.
Laurence des Carrs submitted her resignation to President Emmanel Macron, who said it was "an act of responsibility".
A catalogue of incidents at the world's most visited museum have placed museum bosses come under increasing scrutiny.
The Paris museum issues the first pictures of the crown since the theft, saying it was left "badly deformed".
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The strike action marks the latest disruption for the museum after a jewel theft and water damage.
The leak in the Egyptian department comes just weeks after a jewellery heist that put the museum's security infrastructure under scrutiny.
The price increase is expected to raise millions to fund the museum's renovation plans.
Two men and two women from Paris are the latest to be questioned as part of the ongoing investigation.
