What you have could be called an multivariate logistic-normal distribution, but it is not the same as the multivariate distribution with that name discussed in https://en.wikipedia.org/wiki/Logit-normal_distribution If $X$ is multinormal, the multivariate logit normal distribution we discuss here is the distribution of $Y$, where each component of $Y$ is the logistic transform of the corresponding component of $X$,
$$
Y_i = l(X_i) = \frac{e^{X_i}}{1+e^{X_i}}= \frac1{1+e^{-X_i}}
$$
By calculating the relevant jacobian (which is simple since in this case is the determinant of a diagonal matrix) we can find the density of $Y$ as
$$
f(y)=\frac{\exp\left(-\frac12 (\log\frac{y}{1-y}-\mu_1)^T\Sigma^{-1}(\log\frac{y}{1-y}-\mu_2) \right)}{(2\pi)^{d/2}|\Sigma |^{1/2}\prod_{j=1}^d(y_j(1-y_j))}
$$
where the parameters are the parameters of the multinormal distribution, $d$ is the dimension of $X$ and $Y$, and the log function is applied componentwise. A direct answer to your questions would be easy if we could find the moments in explicit form, but that seems impossible. Let us have a look at an example of this density in the bivariate case, with

$\mu_1=\mu_2=0, \sigma_1=\sigma_2=1$ and $\rho=0.5$. This case is exchangeable, which shows in that the density is symmetric around the line $y_1=y_2$. One way to investigate the correlation is by numerical integration, and for this symmetric case, but with varying $\rho$, this is done below, and the result is that the correlation is remarkably little influenced by the transformation, so to a good approximation the correlation in $Y$ is the same as that in $X$. Some R code:
mu <- c(0, 0)
sigma1 <- 1.0
sigma2 <- 1.0
rho <- 0.5
dmvlogist_norm <- function(x, y, mu, sigma1, sigma2, rho) {
const <- 2*pi*sigma1*sigma2*sqrt(1-rho^2)*y*x*(1-y)*(1-x)
d <- exp(-(1/(2*(1-rho^2)))*( ((log(x/(1-x))-mu[1])/sigma1)^2 +
((log(y/(1-y))-mu[2])/sigma2)^2 -
2*rho*(log(x/(1-x))-mu[1])*
(log(y/(1-y))-mu[2])/(sigma1*sigma2) ) )/const
return( d )
}
# Code for the contour plot:
x <- seq(0.0001, 0.9999, length=101)
xy <- as.matrix(expand.grid(x, x))
d <- dmvlogist_norm(xy[, 1], xy[, 2], mu, sigma1, sigma2, rho)
d <- matrix(d, 101, 101)
image(x, x, d)
contour(x, x, d, nlevels=20, add=TRUE)
title("bivariate logit normal density")
Since there is little hope of finding the correlation via symbolic integration, we use numerical methods:
library(pracma)
calc_corr <- function(rho) {# we can simplify code using exchangeability
m1 <- m2 <- integral2(function(x, y) x*dmvlogist_norm(x, y, c(0, 0),
1, 1, rho), 0, 1, 0, 1)$Q
s1 <- s2 <- integral2(function(x, y) (x-m1)^2*dmvlogist_norm(x, y, c(0, 0),
1, 1, rho), 0, 1, 0, 1)$Q
s12 <- integral2(function(x, y) (x-m1)*(y-m2)*dmvlogist_norm(x, y, c(0, 0),
1, 1, rho), 0, 1, 0, 1)$Q
rhoout <- s12/s1 # using exchangeability
rhoout
}
rho <- seq(-0.99, 0.99, length=31)
rho_logist <- rho
for (r in seq(along=rho)) rho_logist[r] <- calc_corr(rho[r])
The results as a table:
> cbind(rho, rho_logist, rho_logist-rho)
rho rho_logist
[1,] -0.990 -9.320447e-01 5.795527e-02
[2,] -0.924 -9.218907e-01 2.109328e-03
[3,] -0.858 -8.541739e-01 3.826148e-03
[4,] -0.792 -7.871561e-01 4.843921e-03
[5,] -0.726 -7.203866e-01 5.613378e-03
[6,] -0.660 -6.539386e-01 6.061385e-03
[7,] -0.594 -5.877733e-01 6.226669e-03
[8,] -0.528 -5.218579e-01 6.142147e-03
[9,] -0.462 -4.561610e-01 5.838953e-03
[10,] -0.396 -3.906522e-01 5.347779e-03
[11,] -0.330 -3.253032e-01 4.696777e-03
[12,] -0.264 -2.600864e-01 3.913566e-03
[13,] -0.198 -1.949740e-01 3.025960e-03
[14,] -0.132 -1.299396e-01 2.060400e-03
[15,] -0.066 -6.495576e-02 1.044244e-03
[16,] 0.000 3.711409e-08 3.711409e-08
[17,] 0.066 6.495857e-02 -1.041433e-03
[18,] 0.132 1.299417e-01 -2.058303e-03
[19,] 0.198 1.949761e-01 -3.023938e-03
[20,] 0.264 2.600891e-01 -3.910861e-03
[21,] 0.330 3.253074e-01 -4.692627e-03
[22,] 0.396 3.906578e-01 -5.342230e-03
[23,] 0.462 4.561665e-01 -5.833508e-03
[24,] 0.528 5.218657e-01 -6.134344e-03
[25,] 0.594 5.877817e-01 -6.218258e-03
[26,] 0.660 6.539465e-01 -6.053502e-03
[27,] 0.726 7.203924e-01 -5.607636e-03
[28,] 0.792 7.871625e-01 -4.837515e-03
[29,] 0.858 8.542809e-01 -3.719148e-03
[30,] 0.924 9.218908e-01 -2.109192e-03
[31,] 0.990 9.909956e-01 9.956272e-04
In the third column the difference which is mainly in the third decimal.