More precisely, does there exist a sequence G1 < G2 < ... of finite groups such that the irreducible representations of Gn are parameterized by the plane partitions of total size n?
Not if you want the direct analogue of the branching rule to hold: namely, if V is the representation of Gn corresponding to a plane partition A of n, then the restriction of V to Gn-1 is the direct sum of one copy of the representation corresponding to each plane partition of n-1 contained in A. That would allow you to compute the dimension of the representation corresponding to A as the number of paths in the containment poset of plane partitions from the empty partition to A. Some computation then shows that the order of G3 would be 1+4+4+1+4+1=15, but there's only one group of order 15, the abelian one, which doesn't work.
You could imagine some variations of the branching rule, though, such as "if B is obtained from A by replacing k by k-1 then the irrep corresponding to A contains k copies of the irrep corresponding to B", and maybe something like that would work.
I take it this is supposed to be by analogy with the representation theory of the symmetric group? (But it may not be too useful to point this out; anybody who doesn't recognize that isn't going to be able to help.)