This goes and modifies the displayed box form. Please note that this is not meant to be used as input to other computations, just for display - so it's similar to MatrixForm in that sense.
ClearAll[ShortIntegerForm];
ShortIntegerForm[expr_] :=
ToBoxes@expr /.
n_String /;
StringMatchQ[n, Repeated[DigitCharacter, {4, \[Infinity]}]] :>
Tooltip[StringJoin[StringTake[n, 3],
"\[InvisibleSpace]\[Ellipsis]"], ToExpression@n] // DisplayForm
Now you can apply this form to your expression:
(323909701210368 Sqrt[3] t^(56/3))/(11 Gamma[2/3] Gamma[
59/3]) + (23266815064996478976000 t^(71/3))/(Gamma[2/3] Gamma[
74/3]) // ShortIntegerForm

As an added bonus, Tooltip shows the full value if you take your mouse pointer over the short form.