What's the (or some of the) minimum(s) amount of atmospheric carbon dioxide needed by plants?
The values provided in the other answer are actually related to the minimum values experienced by land plants in the past, not the theoretical minimum possible concentration of CO2 capable of maintaining plant life.

According to Gerhart and Ward (2010):


  During the Last Glacial Maximum (LGM; 18 000–20 000 yr ago) and previous
  glacial periods, atmospheric [CO2] dropped to 180–190 ppm, which is among the
  lowest concentrations that occurred during the evolution of land plants.


Thus, what is the theoretical minimum possible concentration of CO2 capable of maintaining plant life?

According to the same paper (Gerhart and Ward, 2010), there has been several experimental studies focusing in low concentration of CO2:


  In the mid-1990s, a later phase of technology was introduced at the Grassland, Soil, and Water Research Laboratory (USDA-Agricultural Research Service) that involves an outdoor tunnel system, whereby plants are grown across a continuum of modern to low [CO2] (Mayeux et al., 1993; Fig. 3b,c). During the day, air of known [CO2] is pumped into one end of the tunnel where plants experience relatively high [CO2]. Plants near the end of the tunnel, on the other hand, experience low [CO2] (similar to glacial values) as a result of photosynthetic removal of CO2 from air as it moves progressively through the tunnel. At night, airflow is reversed while plants are solely respiring, and this serves to equalize [CO2] throughout the tunnel. 


Despite those studies not focusing explicitly on what's the minimum possible concentration, we can still draw some conclusions based on them. For instance, in Fig.4 (ibidem):



As you can see, those vertical lines represent the minimum concentration experienced in the past. However, we still observe CO2 assimilation (photosynthesis) below those values. Unfortunately we cannot simply get the value where the line touches the x axis, because those values are intracellular CO2 concentration. 

That being said, the answer depends on other concept: CO2 compensation point. For a C3 plant, carbon dioxide compensation point is around 50ppm (Tolbert, Benker and Beck, 1995).

Thus, we can say that, as a rough estimate, the value you want lies between 50ppm and 170ppm (probably closer to 170ppm than to 50ppm).

Of course, that minimum possible value depends on several factors, as the species of the plant, the type of photosynthesis (C3, C4, CAM), temperature, humidity etc.

Sources: 


Gerhart, L. and Ward, J. (2010). Plant responses to low [CO2] of the past. New Phytologist, 188(3), pp.674-695.
Tolbert, N., Benker, C. and Beck, E. (1995). The oxygen and carbon dioxide compensation points of C3 plants: possible role in regulating atmospheric oxygen. Proceedings of the National Academy of Sciences, 92(24), pp.11230-11233.