I just came across the TLCA List of Open Problems, which includes 20 unsolved problems in the areas of:
Typed and untyped lambda-calculi as models of computation.
Proof-theory: Natural deduction, sequent calculi, cut elimination and normalization. Propositions as types, linear logic and proof nets.
Semantics: Denotational semantics, game semantics, realizability, categorical models.
Programming languages: Foundations of functional and object-oriented [...]