Is increase in atmospheric $\mathrm{CO}_2$ concentration affecting Earth's atmospheric pressure?
The air in the atmosphere is a fluid. The pressure exerted by a fluid on a horizontal surface can be calculated using the formula.

$$ P = \rho\,\mathrm{g}\, h $$

Where:


$P$ is pressure
$\rho$ is the density of the fluid
$\mathrm{g}$ is the acceleration due to gravity
$h$ is the height of the fluid


The acceleration due to gravity is a constant and the height of atmosphere can be considered constant. That leaves only the density of the air in the  atmosphere as the only variable that will affect pressure on the surface of the Earth, in this situation.

The amount of carbon dioxide in the atmosphere is around 400 parts per million — a very small proportion.

Air density varies according to temperature and pressure but is within the range of $1.146$ to $1.422\ \mathrm{kg/m}^3$. By comparison, water has a density of $1000\ \mathrm{kg/m}^3$.

If the quantity of carbon dioxide were to increase, even tenfold, it would have a negligible affect on the density of the air and thus a negligible affect on the atmospheric pressure on the surface of the Earth. Its effect on temperature is another matter.



Edit

Answer to the additional question: Can you expound on what effect the increase in temperature will have on pressure?

The relationship between pressure, temperature and volume for gases is given by the universal gas law which states,

$$ PV = m\, \mathrm{R}\, T $$

Where:


$P$ is pressure
$V$ is volume
$m$ is mass
$\mathrm{R}$ is the universal gas constant
$T$ is temperature


Rearranging the equation,

$$ P = \frac{m\mathrm{R}T}{V} $$

When considering the affect of temperature on atmospheric pressure, $\mathrm{R}$ is already a constant and the mass and volume of the atmosphere can be considered to be constant because any changes due to increased $\ce{CO_2}$ will be exceedingly small and can be regarded as being negligible.

Consequently $m\mathrm{R}/V$ is a constant $\mathrm{k}$ and the equation becomes,

$$ P = \mathrm{k}T $$

The Kelvin temperature scale, used in the equation, is an extrapolation of the Celsius temperature scale, thus a rise in temperature of $1^\circ\mathrm{C}$ ($1\ \mathrm{K}$) will increase atmospheric pressure by $1\ \mathrm{pascal}$.

The average atmospheric pressure is $101\,325\ \mathrm{Pa}$. Assuming a very bad case scenario of a rise of $10^\circ\mathrm{C}$, the pressure would increase $10\ \mathrm{Pa}$, or $0.0099\%$, which is insignificant.

Weather reports report atmospheric pressure in units of hectopascals ($\mathrm{hPa}$). A pressure of $101\,325\ \mathrm{Pa}$ is $1013.25\ \mathrm{hPa}$. A rise of $10\ \mathrm{Pa}$ is the same as $0.1\ \mathrm{hPa}$. Atmospheric pressures vary between $890\ \mathrm{hPa}$ and $1090\ \mathrm{hPa}$.