For reasons of Economy & Parsimony, The Minimalist Program discards Deep Structure (DS) because it’s not a linguistic level. The main reasons for eliminating DS were based on ‘conceptual grounds’ (related to methodological considerations) not ‘empirical grounds’, at least as far as I know. Is there any ‘empirical reasons’ for eliminating DS, in terms of 'data'?
There is empirical evidence against DS, though it has nothing to do with MP. DS is required for transformationalist theories to move things around so as to get constituents where they are observed to be in surface structures. We need to describe movement, to describe the relationship between active-passive pairs of structures, for instance, among the many movement constructructions studied in transformational grammar. But Gerald Gazdar and others found a way in the 70s and 80s to describe movement without DS and without transformations.
Aside from simplifying TG radically, Gazdar's method predicts some of the Ross constraints on movement. This, then, is the evidence. The Ross movement constraints are not explained, and probably not even formally describable in Chomsky's original TG theory. However, once transformations are removed, the CSC (Coordinate Structure Constraint), the across-the-board condition on the CSC, and perhaps also the CNPC (Complex NP Constraint) "fall out"; that is, they are predicted. This is empirical evidence against DS.
The CSC is due to a general constraint on coordinate structures that only constituents of the same syntactic category can be coordinated. The across-the-board condition on the CSC has the same cause. See for example Gazdar, G. 1981. “Unbounded dependencies and coordinate structure”. Linguistic Inquiry 12.155-184.
If it were not for the CNPC, using Gazdar's non-transformational analysis, an unbounded number of syntactic categories would be required, but this is not possible in a context free phrase structure grammar. Thus, some version of a CNPC-like constraint is also predicted when transformations are eliminated from TG.