Okay, this happens precisely in the obvious case, namely if all primes dividing the order $|G|$ are invertible in $R$.

To see this, note that $\operatorname{Ext}^*_{R[G]}(R,R)$ is group cohomology of $G$ with coefficients in $R$. If $I_G$ were projective, $I_G\to R[G]$ would be a projective resolution of $R$ and thus the cohomology would be $1$-dimensional. However, the cohomology of $G$ with coefficients in $\mathbb{Z}$ has $p$-torsion in arbitrarily high degrees for all primes dividing $|G|$ (this is a well-known fact about cohomology of finite groups). So by the universal coefficient theorem, the cohomology with $R$-coefficients is unbounded, except if $p$ is invertible in $R$ for each $p$ dividing $|G|$.