Learn Finnish: Tätä aarretta ei liene jätetty laivaan. - This treasure was probably not left on the ship.

This is the negative perfect potential in the passive voice, translating to was probably not left. It is formed with the negative verb ei, the potential form of the auxiliary verb olla stripped of its personal ending (liene), and the passive past participle of the main verb jättää (jätetty).

Why is the object tätä aarretta in the partitive case here?

Because the sentence is negative. In Finnish, the object of a negative sentence is always in the partitive case. If this were a positive sentence, the object of the passive verb would be in the nominative case: Tämä aarre lienee jätetty laivaan.

Why do we use the illative case laivaan (into the ship) instead of the inessive laivassa (in the ship)?

The verb jättää (to leave) is treated as a directional verb in Finnish. The action of leaving something is viewed as moving it to a destination, so it requires a destination case like the illative (into) or allative (onto) rather than a static location case.

Could I just use the word ehkä (maybe) instead of the potential mood here?

Yes, and you usually will! The potential mood (lienee / ei liene) is mostly found in formal writing, literature, and news. In everyday spoken Finnish, it is much more natural to say: Tätä aarretta ei ehkä jätetty laivaan.