This company used to offer ferry connections between Turkey and Lebanon. But, it's not possible to book a crossing with them.
Is there any company currently offering ferry crossings to Lebanon? From anywhere in the Mediterranean?
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This agency (also mentioned by @martin.koeberl and @giorgio) responded to my enquiry.
They are offering tickets for the ferry to Tripoli, from Tasucu, charging 200 USD for a one way. It leaves twice a week to Tripoli (but, they say, only once a week to Tasucu).