Does there exist a sequence of groups whose representation theory is described by plane partitions?

Not if you want the direct analogue of the branching rule to hold: namely, if V is the representation of G_n corresponding to a plane partition A of n, then the restriction of V to G_n-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.