"Besides apples, I do not eat fruit."
Translation:Oltre le mele, io non mangio la frutta.
I'm not sure we can use "inoltre" here. In this case "besides" means "apart from, other than," which could be also "all'infuori di" or "tranne" but not "inoltre".
I hope somebody will provide a better explanations, in the meantime here are some links regarding oltre and inoltre.