Rat Study: Excess Iodine Reduces Testosterone
1) Iodine accumulates in the testis resulting in reactive oxygen species.
2) It stimulates the secretion of corticosterone which inhibits LH release. This may only be a problem in rats corticosterone
3) It causes hypothyroidism.

Rat Study: Iodine supplementation reduces sperm count
"These results indicate that thyroid hormone therapy increases plasma SHBG levels in both normal and hypothyroid patients and that this increase precedes the expected elevation of plasma T in males."

Rat study: Iodine and/or Selenium deficiency impacts on rat reproductive function

Rat study: Potassium Iodide, thyroid and testosterone

"Men with primary hypothyroidism have subnormal responses of luteinizing hormone (LH) to gonadotropin-releasing hormone (GnRH) administration and normal response to human chorionic gonadotropin (hCG)."

iodine also proved to mediate the dehydrogenation of testosterone (testosterone into 4-androstene-3,17-dione)
What does this mean?

Bull Study: Potassium Iodide increases testosterone in bulls