Volume is probably not the best way to think about respiration for most of life on Earth, as it only really applies to animals with lungs (a small proportion of earth's biomass). Instead, it would make more sense to think about rates of respiration in terms of oxygen demands. With that in mind, an average human needs around 0.85 kg of oxygen per day (according to NASA). So, the annual oxygen needs of the human population would found by multiplying 0.85 kg/person/day by 365.25 days/year and 7.5 billion people to get ~2.3e+12 kg of oxygen per year.
For a crude estimate of how long it would take to use up our atmosphere's oxygen at that rate, we could simply divide the total capacity of oxygen in the atmosphere (~1.4e+18 kg) by the annual oxygen demand of the current human population (2.3e+12 kg), to get a rough estimate of 600,000 years for Earth's current population to use up all of Earth's current atmospheric oxygen (note that both calculations require some pretty dubious assumptions, but the problem already seems to start that way if I'm reading it right).
For an even cruder estimate of all life on earth, we can again take the Atmospheric Oxygen capacity (~1.4e+18 kg) and divide by the annual oxygen flux from atmosphere to biosphere (~3.0e+14 kg) to get around ~4700 years. (note that this assumes no flux from the biosphere back into the atmosphere, and also assumes that lithospheric flux is negligible.)
If you want to go further and calculate how long it would take for just animals to use up an atmosphere worth of oxygen, you could start by looking at how Earth's biomass is distributed, and make some more assumptions about the oxygen demand of other animals compared to humans following this same idea that I've used above.
Capacity and Flux estimates came (via wikipedia) from (Walker JC (1980) "The Oxygen Cycle". The Natural Environment and the Biogeochemical Cycles), but there are probably more up-to-date estimates available somewhere. The general idea would be the same.