The short version: no consistent effect on thyroid function tests, a species-specific preclinical signal for C-cell tumours, and a contraindication written to be cautious about it.
Reference intervals for thyroid-stimulating hormone are wide, typically about 0.4 to 4.0 milli-international units per litre, and the diurnal swing within a day is a few tenths.
More usefully, pharmacovigilance databases show reports in this area, as they do for every widely used drug, and disproportionality signals from spontaneous reporting are hypothesis-generating rather than evidence of causation.
The caveat is that a family history of medullary thyroid carcinoma or MEN2 is a genuine contraindication and is not something to reason around on a forum.
Rodent C-cell density is the crux of the translation argument, and it is worth knowing once.