Does the increasing rate of expansion of the universe have any implication on or alter the curvature of the universe?

Yes.

"The curvature of the universe" is an imprecise term, and describing the curvature of a general four-dimensional spacetime takes 20 numbers at every point. But I'll assume that your phrase should mean the Ricci scalar curvature $R$, which is a single number at each point that is a kind of average curvature of spacetime (where the averaging is over various "directions" in which one can slice spacetime). In our universe, this number is uniform throughout space (on the largest scales) but decreasing with time. It is asymptotically approaching a constant, nonzero value.

On the largest scales our universe seems to be highly homogeneous, isotropic, and spatially flat (i.e., it has no spatial curvature, but it does have spacetime curvature). Such a universe is described by a Friedmann metric with $k=0$.

The Ricci scalar curvature of such a universe given by

$$R(t)=6left(frac{ddot{a}(t)}{a(t)}+frac{dot{a}(t)^2}{a(t)^2}right)tag1$$

where $a(t)$ is the dimensionless Friedmann scale factor, which was $0$ at the Big Bang and is $1$ today. The dots mean derivatives with respect to cosmological time $t$, and the units are such that $c=1$.

Beginning about 10 million years after the Big Bang, radiation stopped being important to its expansion rate. A solution to the Friedmann equations that takes into account dark energy, dark matter, and ordinary matter (but not radiation, because it is no longer important) is

$$a(t)=left(frac{1-Omega_Lambda}{Omega_Lambda}right)^{1/3}sinh^{2/3}left(frac32Omega_Lambda^{1/2}H_0 tright)tag2$$

where $H_0$ is the current Hubble constant and $Omega_Lambda$ is the current fraction of the energy density that is dark energy.

Substituting (2) into (1) gives the Ricci scalar curvature as a function of cosmological time,

$$R(t)=3Omega_Lambda H_0^2left[3+coth^2left(frac32Omega_Lambda^{1/2}H_0 tright)right]tag3.$$

This is a monotonically decreasing function. As $ttoinfty$, $Rto 12Omega_Lambda H_0^2$.

The relevant parameter values, derived from fitting observational data of the cosmic microwave background to the Lambda-CDM model, are $H_0=0.069text{ Gyr}^{-1}$ and $Omega_Lambda=0.69$. (The similarity of these values is a numerical coincidence.)

One finds that the current value of $R$ (at $t=13.7text{ Gyr}$) is $0.044text{ Gly}^{-2}$, and it is decreasing to the asymptotic value $0.039text{ Gly}^{-2}$.

Here is a graph, where the horizontal axis is cosmological time in billions of years and the vertical axis is the Ricci scalar curvature in inverse billions of lightyears squared:

The reason that $R$ is approaching a constant value is that the increasingly exponential expansion due to dark energy is causing our universe to approach a four-dimensional deSitter geometry, which has constant spacetime curvature. The spatial curvature remains zero, but the spacetime curvature decreases to a constant.