The earth's atmosphere formed mainly through volcanic outgassing, with a minor contribution from cometary impacts. Tectonic activity may or may not significantly change the input into the atmosphere (the presence of plate margins provides an easy escape route for volcanic gases, but sufficient pressure could force an eruption through solid crust in the absence of plate margins), but it certainly does have an effect on which components are removed from the atmosphere by chemical processes and recycled deep within the earth.

In the geologic carbon cycle, atmospheric carbon dioxide reacts with water to form carbonic acid. This acid reacts with metals in surface rocks to form carbonate minerals, which are insoluble in water and are washed into the oceans. The carbonates settle to the ocean floor, where they are eventually returned to the earth's mantle through subduction. Under heat and pressure, the carbonates break down, releasing CO2 which is returned to the atmosphere in subsequent volcanic eruptions. Without plate tectonics to recycle them, the carbon would accumulate on the ocean floor and never return to the atmosphere as CO2. With less atmospheric CO2, the earth would be significantly cooler than it is, possibly cold enough to be locked in a "snowball earth" state.
Source(s):
http://abyss.uoregon.edu/~js/ast121/lectures/lec14.html
http://earthobservatory.nasa.gov/Features/CarbonCycle/
http://www.physicalgeography.net/fundamentals/7h.html
http://www-eps.harvard.edu/people/faculty/hoffman/snowball_paper.html

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