Linear Genetic Programming (LGP) represents programs as sequences of instructions and has a Directed Acyclic Graph (DAG) dataflow. The results of instructions are stored in registers that can be used as arguments by other instructions. Instructions that are disconnected from the main part of the program are called non-effective instructions, or structural introns. They also appear in other DAG-based GP approaches like Cartesian Genetic Programming (CGP). This paper studies four hypotheses on the role of structural introns: non-effective instructions (1) serve as evolutionary memory, where evolved information is stored and later used in search, (2) preserve population diversity, (3) allow neutral search, where structural introns increase the number of neutral mutations and improve performance, and (4) serve as genetic material to enable program growth. We study different variants of LGP controlling the influence of introns for symbolic regression, classification, and digital circuits problems. We find that there is (1) evolved information in the non-effective instructions that can be reactivated and that (2) structural introns can promote programs with higher effective diversity. However, both effects have no influence on LGP search performance. On the other hand, allowing mutations to not only be applied to effective but also to noneffective instructions (3) increases the rate of neutral mutations and (4) contributes to program growth by making use of the genetic material available as structural introns. This comes along with a significant increase of LGP performance, which makes structural introns important for LGP.