Suppose $$ z_i \sim Bernoulli (p_i) $$
Can we use CLT for the following weighted sum?
$$ S = \sum_i w_i z_i $$
i.e. can $S$ be approximated with a normal distribution? If yes, with which theorem? (I suppose the classical CLT holds only for average of iid variables)
Either the Lyapunov CLT or the Lindeberg CLT will be what you seek.
In each case let $X_i\,=\,w_i\,z_i$ and apply the theorems as given there to the $X_i$.
In a great many cases of the kind you suggest (and likely all that you care about), checking Lyapunov's condition should be sufficient.
However, unless you have some weird edge case, I think Lindeberg's should work in all cases of the kind you need.
