Independent copula vs Student-$t$ copula with zero correlation matrix? Suppose I have the random variables $X_1, \dots, X_n$ with the marginal distributions are not normal (in fact, unknown marginal distribution).
Will there be any difference between the assumption $X_1, \dots, X_n$ are independent and $X_1, \dots, X_n$ are modeled with Student-$t$ copula with the correlation matrix is all zeros (with $1$ on the diagonal)?

The uncorrelated $t$ copula is not the same as the independence copula. It is based on the multivariate $t$-distribution, which is an elliptical family, and the only elliptical distribution for which zero correlation implies independence is the normal. The difference can be quite large.
Below we will illustrate this using the R package copula. A contour plot of a $t$-copula is

The density of the independence copula is a constant 1. Note how the $t$-copula concentrates probability in the center and close the the four corners. The code used is
library(copula)
indCop <- ellipCopula(family="normal", param=0, dim=2, dispst="ex")
tCop <- ellipCopula(family="t", dim=2, dispst="ex", param=0, df=2)
 getSigma(indCop)
     [,1] [,2]
[1,]    1    0
[2,]    0    1
getSigma(tCop)
     [,1] [,2]
[1,]    1    0
[2,]    0    1

# See they are different:
dCopula(c(0.5, 0.5), indCop)
[1] 1
dCopula(c(0.5, 0.5), tCop)
[1] 1.27324

contour(tCop, dCopula, n.grid=101, levels=c(0.5, 0.6, 0.7, 0.8, 0.9, 1, 1.1, 1.2, 1.3), main="t copula, uncorrelated, df=2")

