Asking for recommendation letter before beginning internship
You're hugely overthinking this.


It's normal for an undergraduate project student to ask their supervisor for references
It's normal to be making applications, and requesting references, before your course of study is complete


Consequently, your future supervisor knows that he may well be asked to provide a reference for you and doens't need "warning" of that fact now. Asking him now won't cause him to increase your workload, although it's unlikely to do any harm (context: OP seems to consider an increased workload to be good) either.

That said, once you know what you're applying for, you should give him details and ask him if he is willing to provide a reference well in advance of any deadlines. This will give him time to prepare a reference letter (or whatever's required) and will enable him to tailor it to your specific applications; it will also prevent him being caught unprepared if they just phone him and ask him about you.