How does a polymerized TruSeq adapter affect sequencing reads?
In short, you stand to lose information from your library due to the length of the fragments you're sequencing. 

When you set up your sequencing run, you'll set the parameter for the number of sequencing cycles, which will determine the length of the reads to be sequenced. The longer the fragments contained in your library, the less they'll be fully sequenced. 

A good idea would be to sequence your suspected polymerized fragments to verify that primer polymerization is really occurring. Just send your sample for normal sequencing, as you would for plasmids and PCR fragments. 

If your primers are polymerizing because of inefficient design, you should fix this before loading your library into the Illumina device. If the primer-dimer effect comprises a significant portion of your library, this will only get amplified during the run and can definitely lead to a significantly less complex library. Sequencing a low-complexity library is a really expensive mistake whose one guaranteed result is to enrage your advisor and make him/her loudly question your competence. Trust me.