To pick the incorrect statement, recall the genetics of toxigenicity in Corynebacterium diphtheriae. The capacity to make diphtheria toxin depends on the bacterium being infected (lysogenised) by a temperate corynephage that carries the $tox$ gene. Therefore the toxin gene is phage-encoded, not located on the bacterial chromosome, which makes the claim of chromosome mediated toxin production wrong.
Reviewing the other choices confirms they are all correct. The clinical disease results from the secreted exotoxin rather than tissue invasion, so damage by toxin is true. The toxin acts by ADP-ribosylating elongation factor-2, halting protein synthesis, with the heart (toxic myocarditis) and peripheral and cranial nerves (post-diphtheritic paralysis) being the chief targets, so cardiac and neuronal toxicity is true. Epidemiologically the infection predominantly affects non-immunised young children, so that statement is true as well.
The one factually wrong option is that toxin production is chromosome mediated.\[\boxed{\text{Toxin production is chromosome mediated (it is phage/beta-corynephage mediated)}}\]