That’s a complex question, and the short answer is “yes and no”. Yes, is transportation involved in bringing the CO2 to wholesalers like Tavern Service or when when they bring CO2 to restaurants. Same with all other products that restaurants and bars use. CO2 is like their other food and supplies in that way. A truck had to get the gas to them and that truck runs on gasoline or diesel which produce CO2 when the fuel is burned.
But no, CO2 is just a by-product of the oil refining process. One of the raw ingredients used in the refining process is hydrogen. The hydrogen is injected into the crude oil in refining which helps produce higher quality gasoline and other fuels. When hydrogen is injected into the oil in the columns, CO2 that is already in the crude oil is freed. Most of that CO2 is just released into the atmosphere, but a small amount is captured for beverage and industrial use. It is sold to the big gas distributors like Air Liquide, Praxair, and Linde.
Those big gas distributors then greatly refine and filter this raw CO2 and use it to produce several grades of it. Beverage grade is the most common form and what we use at Tavern Service. It is 99.90% pure and free of benzene and any other petroleum by products. All beverage distributors use it, and it is how all soda and most beer manufacturers carbonate beverages. (The exception is that a few large beer brewers like Sierra Nevada, New Belgium, Alaskan Brewing, and some Anheuser-Busch locations capture the CO2 that the yeast produces in the fermentation process, filter it, and inject it back into the finished beer).
The bottom line is nothing is burnt in producing the CO2 that is used in the beverage industry, it is a waste product that otherwise would be vented into the air. It all comes from recycled sources. But some CO2 is produced in getting the gas to the final user.