Qatar has since the eruption of the Gaza war last October played an essential role as a mediator between Hamas and Israel. But if Doha cannot convince Hamas to release Israeli hostages seized on Oct. 7, 2023 in the coming weeks, it might have to reconsider its hosting of the Palestinian group’s political office.

In Nov. 2023, Qatari mediation efforts, alongside those of other countries, resulted in the freeing of more than 100 captives held in Gaza—including 80 Israelis. This was immediately followed by a seven-day truce. Subsequently, Doha facilitated another breakthrough in January this year, when it coordinated with...