Ice chunks would be melted off during closer approaches, potentially blasting non-ice bits off of the exocomet as it goes. This could affect the trajectory of the exocomet both by the inertial force of the separation and because the mass would be reduced.
Eventually it would run out of water, which means it would no longer be a comet--the streaming tail of gas wouldn't exist anymore, even on closer approaches to the sun. What would remain--assuming anything remained at all--would be a hard rocky core.
From there, the orbit would either stabilize or decay to a point that it crashed into the sun. Even a fairly stable orbit could eventually bring it close to other gravitational bodies in the solar system which could throw it into an unstable orbit, or alternatively, it could collide with a planet or asteroid.
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