Here is a paper and summary on this topic: Essentially even though the study was on preterm infants the level of thyroxine in breast milk is very low and won't affect the infant significantly in that way.
Clin Endocrinol (Oxf). 2002 May;56(5):621-7.
The quantity of thyroid hormone in human milk is too low to influence plasma thyroid hormone levels in the very preterm infant.
van Wassenaer AG, Stulp MR, Valianpour F, Tamminga P, Ris Stalpers C, de Randamie JS, van Beusekom C, de Vijlder JJ.
Thyroid hormone is crucial for brain development during foetal and neonatal life. In very preterm infants, transient low levels of plasma T4 and T3 are commonly found, a phenomenon referred to as transient hypothyroxinaemia of prematurity. We investigated whether breast milk is a substantial resource of thyroid hormone for very preterm neonates and can alleviate transient hypothyroxinaemia. Both the influence of breast feeding on plasma thyroid hormone levels and the thyroid hormone concentration in preterm human milk were studied.
Two groups were formed from the placebo group of a randomized thyroxine supplementation trial in infants born at < 30 weeks' gestational age on the basis of the mean breast milk intake during the third, fourth and fifth weeks of life. One group received more than 50% breast milk (mean breast milk intake 84%, n = 32) and the other group less than 25% breast milk (mean breast milk intake 3.3%, n = 25). Plasma thyroid hormone concentrations were compared between the two groups. Breast milk was collected from mothers of infants participating in the same trial and the thyroxine concentration in breast milk was measured with RIA after extraction.
No significant differences were found between both groups in plasma concentrations of T4, free T4, T3, TSH, rT3 and thyroxine-binding globulin (TBG), which were measured once a week. Thyroxine concentration in breast milk ranged between 0.17 microg/l and 1.83 microg/l (mean 0.83, SD 0.3 microg/l) resulting in a maximum T4 supply of 0.3 microg/kg via ingested breast milk. In formula milk, the T4 concentration was equally low. Protease treatment did not influence the measured T4 concentrations.
No differences in plasma thyroid hormone between breast milk-fed and formula-fed infants were found. The amount of T4 present in human milk and formula milk is too low to alter the hypothyroxinaemic state of preterm infants.