At the present time, volcanoes on the Earth inject new gasses into the Earth's atmosphere; primarily carbon dioxide and sulphur-rich compounds. Now carbon dioxide is a so-called greenhouse gas because in the atmosphere it allows optical sunlight to pass to the ground, but does not let the longer-wavelength infrared light produced by the heating of the Earth's surface to escape back out into space. It is this 'trace component' to the Earth's atmosphere which controls how hot the surface of our planet is, or will be, not the total quantity of gas in the atmosphere which is dominated by nitrogen and oxygen and which are gases not exhaled by volcanoes.
Atmospheric scientists are just beginning to understand the carbon dioxide cycle; where it comes from and how it is lost from the atmosphere. Apparently, the oceans are a good 'sink' for carbon dioxide and are capable of burying millions of tons of CO2 each day by first carbonating the water, and then burying the carbonation in the deep ocean.
The Earth is loosing some of its atmosphere into space because even though most of the air atoms are moving well below the Earth's escape velocity, way up in the outer stratosphere, the atoms get heated by their interaction with sunlight and other energy sources. Some very small fraction of these atoms do achieve escape velocity and are lost from the Earth. Can the Moon enhance this process? If the gravitational force of the Moon and Earth are combined, their ought to be a region on the line joining the centers of the Earth and Moon where the effective gravitational force felt by a atom is slightly lower than if the Earth alone were present. The escape velocity at this point would be slightly lower than elsewhere, and this could enhance the escape of gas atoms in the upper stratosphere. But, this slight trickle of gas away from the Earth would have little effect on the heavier gases and probably only affect the lighter atoms such as hydrogen and helium. As for the impact of this leakage on the greenhouse effect, it is probably negligable because CO2 molecules are trapped in the lower atmosphere because they are very heavy; and the sources and sinks of CO2 are dominated by the oceans and biomass near the Earth's surface which have far greater consequences.