Ok, just so happens I am a chemistry major. The gas dispels oxygen because carbon dioxide is heavier than oxygen (higher molecular weight) I won't go into technicalities because I am not sure how much you know about chemistry. 44 grams of dry ice will become a gas, at very roughly 0.022 m^3, or 220 cubic centimeters. It is a bit of math, but you can find the volume of whatever container you are using, let's take a 5 gallon bucket.
Measure the height of the cylinder, the radius( measure halfway across the bottom at the widest point of the circle). The volume is the height x the radius squared x 3.14( or you can just google it. Fill the container and guestimate how much volume of air is left.
I rambled a bit, but hopefully this info was useful.
Almost forgot, for a solid 28grams=1 oz
Dude, you are giving us chemists a bad name. You really want to figure out the
volume of a 5 gallon bucket by multiplying its height by radius squared by pi? And you did not answer the question pose.
5 galon bucket is about 19 L, or roughly the 0.022 m3 (the aprx volume of 1 mole of gas at STP), which trainedtosave mentioned. 44 grams (i.e., 1 mole) of CO2, will just about fill up that bucket. Given that you need a bit more in order to flush it, given that CO2 (44 g/mol) flushes air (~29 g/mol) rather well if the dry ice is on the bottom and the escape is at the top, that some space is taken up by the foodstuff, and that you'll need the dry ice to generate about 2 or 3 times the volume of gas to flush all of the air out well, I'd say that:
(1) use 2 to 5 oz, of dry ice per 5 gal bucket (less then 2 wont flush air out, more than 5 will be wasteful); and
(2) you must let the escaping gasses out of the container through a hole as tiny (say 1/16") as possible.
As soon as the gasses stop escaping, just plug up the pinprick hole. Burping the lid will introduce too much air into the container.
PS: Assignment for trainedtosave: just how many cubic cm are there in 0.022 m^3?

: Extra credit: How much O2, in percent of total volume, will there be if Maverick drops 5 oz of CO2, into 5 gallon bucket filled with rice? Assume kB = 1.38×10^-23 J/K, 1/16 inch effusion orifice, first order sublimation of CO2 with half life of 1 hour, each rice grain is a sphere with a 2 mm diameter, and cubic closest packing.