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To get a ZF-definable function that is not Lebesgue integrable on any interval, we need a function that is measurable but has \int_I |f| = \infty for every interval I.

Example: Define

f(x)=
\begin{cases}
\infty, & x\in A,\\
0, & x\notin A,
\end{cases}

where A is a measurable set of infinite measure in every interval. But such a set must have full measure in every interval, so A=\mathbb{R} a.e., and then f is essentially the constant \infty, not real-valued.