While no particular Venn diagram can constitute a proof, I think that a pair of trees of Venn diagrams surely can. For example, to prove the identity $$A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$$ you could construct two trees, one for the LHS, the other for the RHS. The nodes of these trees are themselves Venn diagrams. Consider, for example, the LHS tree. It has three leaves, which are Venn diagrams denoting $A,B$ and $C$ respectively. The nodes $B$ and $C$ are joined to a third node $B \cup C$ above them, which is a Venn diagram denoting their union. We then join the nodes $A$ and $B \cup C$ to another node above them, which is a Venn diagram denoting their intersection. Construct a tree similarly for the RHS. Since the topmost Venn diagrams "look identical," this proves the identity of interest.
This is essentially a pictorial way of doing Truth Tables. The main downside is that it is limited to proving identities involving three variables.