Are there some theorems about the splitting of infinite conjugacy classes into several conjugacy classes in a subgroup? I am mainly interested in subgroups of finite index. Thanks.

I'm not sure what you mean by an infinite length conjugacy class. Most likely you mean that the cardinality is infinite. Consider the Heisenberg group generated by 3 elements $x,y$ and $z$ with relations so that $z$ is central and $xyx^{1}y^{1}=z$. Then the conjugacy class containing $y$ consists of all elements of the form $yz^n$ for integers $n$. If we pass to the finite index subgroup generated by $x^k,y$ and $z$ for some natural number $k$ this splits into $k$ distinct classes represented by $yz^i$ for $i=0,\ldots k1$. 

