Non-classical logics usually don't have a problem with the deduction theorem, as long as they have no relevancy based implication, i.e. if they are based on a residuate lattices.
Many people on the other hand believe that the deduction theorem does not hold in modal logics, especially not in interesting logics such as temporal logic. A typical argument goes as follows. In modal logic we would have an inference rule:
P
----
[] P
And therefore if a deduction theorem would be available, we could proof P -> [] P, which is not desired. This argument is for example informally repeated in Temporal Logic, The Lesser of Three Evils, Leslie Lamport, Microsoft Research, MSR-TR-2004-104.
Fortunately matters are not that worse. A more detailed analysis is given by Does the deduction theorem fail for modal logic? Raul Hakli, Sara Negri, November 10, 2010. In a Hilbert Style calculus HK the above rule should be more precisely formulate as follows:
|- A
---------
G |- [] A
The deduction theorem then holds. Since we cannot prove |- P in the first place, and therefore also not go to |- P -> [] P. Besides a Hilbert Style calculus, the paper also presents an equivalent Gentzen Style calculus which has the deduction theorem as an inference rule.
Bye