I'm not sure Margaret Hodge, the person who ignored child abuse victims whilst leading Islington Council and libelled one of the victims, is the best person to comment. In any case, she needs to check her facts. Some of what she said is wrong.
Andrew paid £1M for a 75-year lease on Royal Lodge and was required to spend £7.5 million on refurbishments. This means he has effectively paid £113,333 rent per year, although if we were doing a proper conversion to an annual rent the figure would be somewhat higher due to the fact he has paid up front. He may have paid below the market rate for the property, but he is definitely not living rent free.
Queen Elizabeth's estate was not subject to IHT, but Andrew's will be. Her estate did go through probate, but royal wills are sealed, usually for 90 years. Contrary to what many people think, royal wills are not the only ones sealed. It doesn't happen very often, but a judge can seal any will if they deem it undesirable or inappropriate for it to become public, e.g. to protect personal safety or sensitive information. We don't know whether Andrews will is going to be sealed.
Andrew does not receive any income from the Duchy of Lancaster or the Duchy of Cornwall. Those who do (Charles and William respectively) voluntarily pay tax on their income. The accounts of both Duchies are in the public domain.
The only public money Andrew receives is his Armed Forces pension.