1) If you know what is your compound. In this case you can calculate the amount of carbon oxide evolved using simple stoichiometry involved in the oxidation reaction with oxygen.
CxHxNz + xO2 --> xCO2 + other compounds
In other words, the number of moles of evolved CO2 is equal the number of moles present in the organic molecule. From the molar value you can calculate the mass of evolved CO2.
2) If you don't know your compound. You can perform an CHN elemental analysis. Were you will get the %C of your organic compound. From this value you can get again the amount of CO2 evolved.
(Redfield AC (1934) On the proportion of organic derivatives in sea water and their relation to the composition of plankton. James Johnstone Memorial Volume University Press, Liverpool. P.176-192.)
You can trap the CO2 released in a closed vessel, take sample with syringe, inject it into Infra Red Gas Analyser (IRGA), which would give you the CO2 concentration in ppm
Atul is right. If you have access to a total organic carbon analyzer, you can measure TOC before and after the reaction. The missing amount is inorganic C (virtually all CO2). No carbon analyzer? You'll need to do something like a chemical oxygen demand test before and after and then back calculate carbon loss from stoichiometry. Trapping the CO2 is tricky with the Ba(OH)2 method unless you are very good at catching all of it. That means all the gas evolved during your reaction must make it to the Ba(OH)2 solution, and at the end you'll have to purge very efficiently (add acid first to produce all carbonates as CO2). My low tech approach would be the COD route - kind of toxic though, with disposal being a slight issue.
It would depend on whether you are interested in mineralization by aerobic or anaerobic routes. But EPA has guidelines for this kind of work that can be found on the internet or on their web site www.epa.gov. The Guidelines that you might find particularly useful are Aerobic soil metabolism (OPPTS 835.4100) Anaerobic soil metabolism (OPPTS 835.4200); Aerobic Aquatic Metabolism (OPPTS 835.4300) and Anaerobic Aquatic Metabolism (OPPTS 835.4400). There are other experimental methods that also do this that can be found under OPPTS series 835 and in the OECD Guidelines: OECD Guidelines 307 and 308. While the OPPTS guidelines are available on-line, the OECD guidelines can be purchase from OECD but don't appear to be available on line.