All the textbooks I saw are very clear about the implications of commutating operators in quantum mechanics. However, much less is said about anti-commutation relations. Does it have a general implication if two operators anti-commutate?
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1) For fermionic operators, the important object is the anti-commutator, not the commutator, see e.g. this question.
2) However, since OP did not mention fermionic operators explicitly, it seems that OP is asking of the role of anticommutators for bosonic fields only.
3) In the case of uncertainty relations, see e.g. this question.
4) Finally, it should also be mentioned that the defining relation of a Clifford algebra uses an anti-commutator.
