Does Povidone-Iodine that penetrates through the skin stays in the body (cells, liver etc)?
Your source (DrugBank via PubChem) is describing the typical pharmacokinetic information (ADME, absorption, distribution, metabolism, excretion) on Providine-Iodine, under the assumption it is not absorbed. It's true, a thing that is not absorbed has no volume of distribution, is not metabolized, and is not eliminated.

Though Providine-Iodine is not absorbed as is, Iodine is.

From the Lexicomp drug information for prescribers:


  Absorption: Absorbed systemically as iodine; amount depends upon concentration, route of administration, characteristics of skin


Other pharmacokinetic properties are as for Iodine. It may be taken up by the thyroid. Clearance is primarily unchanged (as Iodine) by the kidney.

The absorption through the skin as iodine and elimination by the kidneys are the reasons for the prescribing precautions:


Use with caution in patients with burns (excess absorption)
Use with caution in patients with renal impairment (impaired excretion)
Use with caution in patients with thyroid disorders (taken up by the thyroid)


This case report of Iodine toxicity following prolonged povidone iodine administration is available on pubmed central. It's from the 80s, but it is still relevant and the comments section has a good overview:


  The reports of iodine absorption from topical povidoneiodine
  solution suggest that absorption is enhanced when the
  compound is applied to denuded skin, mucosal surfaces with
  high absorptive capacity or extensive areas of intact skin. The
  evidence for iodine absorption includes documented serum
  iodine concentrations and thyroid function abnormalities.