This is a very vague question.

We know that some algebra structures can be viewed as module of some fantastic stuff, call T. Such example include is Abelian groups are $\mathbb{Z}$-modules, chain complexes are modules of dual numbers $\mathbb{R}^{1|1}$(maybe not so correct, multiplication differs from usual one), more generally modules over a operad T (Lie algebra and associative algebra are of this kind). To get the infinitified version(Lie-infty A-infty, etc.) one use a infinity operad. Note that module is basically a morphism from T to some Auto( ). So the infinitified version replace the source thing by a higher thing.

Projective representation goes differently. A representation is basically the same as a module, a morphism from T to some Auto( ). Projective representation (we could generalize this idea, such as a group representation in a 2-category) basically replaces the target by a higher thing.

A nature question is this two construction the same? It seems obvious not. On the other hand, I feel they should have some relation.