Let \(RG\) be the group ring of the group \(G\) over the associative ring \(R\). For \(n \geq 1\) define \(RG^{[1]}\) to be \(RG\) and for \(n > 1\) to be the two-sided ideal of \(RG\) generated by all left-normed Lie commutators \([x_ 1,x_ 2,\dots,x_ n]\) \((x_ i \in RG)\), where \(ab-ba=[a,b]\). The group ring \(RG\) is said to be Lie nilpotent, if \(RG^{[n]}=0\) for some \(n \geq 1\). In this paper the authors study group rings which are residually Lie nilpotent, i.e. for which \(RG^{[\omega]}=\bigcap_ n RG^{[n]}=(0)\), when \(R\) is a field or \(R=\mathbb{Z}\), the ring of integers.