If the orbit of the Moon was elliptical or in someway different from what it is currently, would that change affect the climate of the Earth?
The most important effect of the Moon on the Earth is tides. The act of transporting huge amounts of water around is important for circulation and exchange of matter, and is also making the shore line a wide and ecological interesting area. If the Moon had another orbit, the tides would have another strength or frequency.
Microbiological life in the top layer of the ocean is responsible for a lot of the photosynthesis, important for the composition of the atmosphere. They are dependant on the transport of nutrients between coastal and open ocean zones.
The Moon may also protect the Earth from meteorites, event though that is disputed.