The "for" is mandatory with "responsible". You can be (just plain) "responsible" without any preposition, but if there is another noun (or pronoun, or other noun-like object) after responsible, you need a preposition there to specify the relationship between that noun and "responsible".
The combination "... for are ..." doesn't sound bad to me in this context. If you really don't like it, you could reword the sentence like this:
The tasks for which this layer is responsible are meaningful to the business or necessary for
interaction with the application layers of
But, to me, it sounded fine the way it was. Either way, you've got a somewhat complicated sentence there.