In the literature, Postnikov systems seem to be defined always in the setting of CW complexes. Looking at the proofs, it is not clear to me, why this assumption should be necessary.
Question: Does there exist a Postnikov system for an arbitrary topological space?
Reminder: a Postnikov system for a space $X$ is a sequence of fibrations $X_n\to X_{n-1}$ such that $\pi_k X_n=0$ for $k>n$ and $\pi_k X_n=\pi_kX$ for $k\le n$. One usually constructs $X_n$ by succesively gluing cells to $X$ to kill all homotopy groups above degree $n$. It is then clear that one has an inclusion $X_{n+1}\to X_n$ which can be replaced by its mapping cylinder (homotopy equivalent to $X_{n+1}$) to make it a fibration.
So the question might be: is there something going wrong with killing homotopy groups in non-CW spaces? I wouldn't think so because the argument (for killing homotopy groups) only uses that one is gluing the pair (disk, sphere) and seems not to make use of any property of $X$.
Migrating my comment to an answer: there is no problem as long as you're willing to work up to weak homotopy equivalence. Any complication that might occur would come from point-set difficulties arising from demanding at some point to work up to homotopy equivalence or worse instead. (For example, already universal covers don't always exist, but "universal covers up to weak equivalence" always do.)