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The irrevocable trust is a separate legal entity and would be a subject of capital gain on the sale of the property.
There is no pass through provision for the irrevocable trust.
There is no primary home exclusion as it would if your father sells the home or the home would be held in a revocable trust.
There would not be a stepped-up basis if the property will be passed from the trust to beneficiaries as it would be for inherited property.
That is very important decision to place the property into the irrevocable trust because such transfer may not be simply revoked and your father would change his mind - the transfer the property back to him would be treated as sale and will be taxable event for the trust.
The irrevocable trust would allow to separate the property from the original owner and your father would be able to qualify for instance - for Medicaid after five years - and at the same time will have a control over the property, but he will loose some tax advantages.
Please see here a brief comparison between Revocable and Irrevocable Trusts -
Let me know if you need any help.