SAT - Hardness of determining backbone literals
The complexity is co-NP complete because you're converting to unsatisfiability of $F \land \lnot l$. I have a proof of the completeness in my PhD thesis, Claim 2 (if I may blatantly advertise myself).

There's a small problem for unsatisfiable formulas, if your definition of a backbone means that all the literals are backbone for an unsatisfiable formula, then the problem stays in co-NP. If, however, the set of backbone literals is empty for an unsatisfiable formula, you also need NP.