Full Idea
The problem with material implication, and classical logic more generally, is that it considers only the truth value of formulas in deciding whether to make an implication stand between them. It ignores everything else.
Gist of Idea
Material implication (and classical logic) considers nothing but truth values for implications
Source
Edwin D. Mares (Negation [2014], 7.1)
Book Reference
'Bloomsbury Companion to Philosophical Logic', ed/tr. Horsten,L/Pettigrew,R [Bloomsbury 2014], p.208
A Reaction
The obvious problem case is conditionals, and relevance is an obvious extra principle that comes to mind.