This is a very long-term change, so what implications there will be will most likely end up being less important than other changes. In particular, the rate of change of orbit is such that the Moon's orbit will have increased by 20 days in about 50 billion years. By which time, the Sun will have been dead for about 46 billion years or so.
Assuming no other changes, the Moon is predicted to reach a "locked" orbit of 47 days about the Earth, at which point both the Earth and the Moon will rotate about their axes at the same rate (so, rather like our only seeing one side of the Moon, people on the Moon would only see one side of us).
It's hard to know how this will change things exactly -- perhaps few people have given proper thought to the situation, as after all by the time there's any noticeable changes it won't matter anyway.