Finally, a Math Overflow question that addresses my specialty: non-rigor!
Here are a few examples of non-rigor as applied to evidence for dualities:
Heterotic-Type II. In earlier times, the best evidence for heterotic-Type-II duality was a) counting the number of supersymmetries of the theory, and (b) comparing the moduli spaces.
AdS-CFT. For AdS-CFT the earliest and best comparisons were counting the so-called anomalous dimensions of various operators. To date, I think the tests are far from rigorized (and yes, this would be a great problem to make mathematically precise).
Mirror Symmetry, early days. Recall that mirror symmetry in CY moduli space came from constructing a chart of the Euler characteristics of CY complete intersections and noticing the symmetry of the chart about zero. Other non-rigorous arguments involve counting the dimensions (just the dimensions) of the moduli of purportedly mirror objects. Then there's the old compute-on-flat-space-and-let-supersymmetry-take-care-of-the-rest trick.
Low energy effective field theory. The "fact" that string theory reduces to an oft-identifiable QFT in a low energy limit is a huge source of argumentation/inspiration in string theory. Accounting for (effective) black holes helped lead to M-theory in one context, and to the microscopic description of black-hole entropy in another. One can also argue for dualities by identifying equivalent field contents in two different models. This brings up another point.
Invariance of BPS states under perturbation. It is great to take a quantity that does not vary and evaluate it in a limit where it is easy to compute. This argument appears again and again in physics -- and also in math, of course (e.g. in the heat-kernel proof of the index theorem). BPS numbers are just that. (Of course, they do vary, and the continuity of the relevant physical parameters [numbers are not necessarily physical quantities] is what underlies interesting explanations of wall-crossing.)
I'm probably including too many that don't fit and excluding a lot that do. Very non-rigorous of me!