The documentation contains the following inverse for the 1D case:

    dtft = ListFourierSequenceTransform[{1/5, 1/4, 1/3, 1/2, 1}, ω]
    Table[InverseFourierSequenceTransform[dtft, ω, n], {n, 0, Length[dtft] - 1}]

In higher dimension

    InverseListFourierSequenceTransform[L_, p_List] :=
             Normal[SparseArray[With[{sz = Length[p]},
               With[{xs = Unique[ConstantArray["x", sz]], 
                     cs = Unique[ConstantArray["c", sz + 1]]},
             (List @@ InverseFourierTransform[L, p, xs + 1, FourierParameters -> {1, 1}] /.
               Pattern[Evaluate[Last[cs]], Blank[]] Times @@ MapThread[DiracDelta[# + 
                 Pattern[#2, Blank[]]] &, {xs, Most[cs]}] -> #) &[Most[cs] -> Last[cs]]]]]]

which also works in 1D if you provide the variable argument in a list.

Note that it is only a right inverse. A left inverse doesn't exist because of the we don't have injectivity:

     ListFourierSequenceTransform[{1/5, 1/4, 1/3, 1/2, 1, 0}, \[Omega]] ===
       ListFourierSequenceTransform[{1/5, 1/4, 1/3, 1/2, 1}, \[Omega]]
>True