Is volume of air increasing as CO2 levels increase?
[Major edits below]

In short the answer is NO.

Before we get into volume changes, I have to say that volume is a tricky measure to apply to the atmosphere as a whole, because there is no clear limit between the atmosphere and outer space.
Also, the thickness of the atmosphere is quite insensitive to changes in atmospheric mass. For example, the atmosphere of Venus is much more massive than Earth's (93 times more massive). Leading to surface pressures 92 times those of Earth. However, the thickness of the atmosphere is quite similar, with the mesosphere extending to 120 km, only 41% higher than on the Earth, despite having 9300% the mass.

This planetary comparison suggests that more CO$_2$ will lead to a more massive atmosphere and higher atmospheric pressures, but little change in its actual volume (if you somehow manage to define how to measure the volume of the atmosphere).

As an analogy, imagine you have one meter of snow cover, and there is one meter of snowfall. You won't end up with two meters of snow cover. Thise is because the new snow will compact the snow below, leading to something like 1.7 meters of snow cover.

Therefore, more gases in the atmosphere will lead to a denser atmosphere, but have little impact on its thickness.

Now we have to make a very important distinction: It is not the same to add CO$_2$ from a CO$_2$ source (like a volcanic eruption) than add it due to combustion of fossil fuels. This is because the latter do not add molecules to the atmosphere, just turn O$_2$ molecules into CO$_2$. Now, if you look into what this transformation means in terms of volume, you will get a surprise (at least it was a surprise to me):

Given standard conditions (20°C and 1 atm). One cubic meter of O$_2$ weights 1.331 kg. When you use that oxygen to burn fossil fuels all the molecules in it will go from a molecular weight of 32 (O$_2$) to 44.01 (CO$_2$), meaning you end up with 1.831 kg of CO$_2$. Now comes the interesting part: under the same conditions, one cubic meter of CO$_2$ weights 1.842 kg, consequently, our initial cubic meter of O$_2$ turned into 0.994 m$^3$ of CO$_2$, equivalent to a volume reduction of 0.6%. Therefore, burning fossil fuels reduces the volume of the atmosphere.

I was surprised at first by this result, but it nicely explain the classic experiment where a boiled egg get sucked into a bottle:

This is an screenshot of this video where you can see the trick in action (skip to minute 1:10)

Now, despite that burning stuff decrease the volume of the atmosphere. It is important to note that if you add greenhouse gases like CO$_2$ or methane, you are not only increasing the amount of those gases in the atmosphere, because the increase in temperature will lead to an increase in water vapor too. So the atmospheric mass increase will be larger than the mass of CO$_2$ or methane added. And that extra mass is related to additional molecules, so it will lead to extra volume (contrary to the combustion case where the amount of molecules is constant).

Thus, as we burn fossil fuels the mass of the atmosphere increases (as well as the atmospheric pressure), but the volume decreases, and due to the fact that CO$_2$ is the product of combustion, that increase will be at the expense of reducing the amount of molecular oxygen (O$_2$).

Although, this decrease is very small: the atmosphere have ~21% of Oxygen and 0.04% of CO$_2$, so an increase in CO$_2$ of a 30% (similar to going from 300 to 400 ppm), means a reduction of O$_2$ of 0.01%, a relatively small change (using Wikipedia numbers it would go from 20.946% Oxygen to 20.934%). Also, that's strictly true only if you make an instantaneous combustion. Because over time there could be negative feedback working against this imbalance. For instance, plants will increase the rate of photosynthesis (CO$_2$ fertilization effect)

That said, the mass increase (and volume decrease) associated to the CO$_2$ jumping from 300 ppm (pre-industrial levels) to 400 ppm is negligible, and probably dwarfed by day to day changes in particulate material or water vapor.

As a side note, the book "Red Mars", makes a fairly scientifically sound description of how to "terraform" Mars, and how adding CO$_2$, or water vapor increase the atmospheric pressure and surface temperature.