I would not give your dad the house, even within an irrevocable trust, because either way it's considered a gift. With an irrevocable trust, your dad would be the beneficiary & you'd be the trustee(s). A trust would shield the house from his creditors but your ownership already does that, so I don't see the point. You'll have to make sure it would qualify for property tax waivers and/or a homestead exemption. If you don't pay the house off, check with the mortgage holder to make sure the house can go into such a trust. If you want the trust to revert to you, either your dad must put this in his will (which is difficult since he no longer can make his own decisions) or the trust document must say so, but that makes it less than an arm's length transaction, which an irrevocable trust must be -- check with an estate attorney, who should be the one to address these issues & draft the proper language for your state. Since a trust is its own tax entity, the trustee has to file a return every year regardless of whether any tax is actually due.
Also look into putting the house into a revocable trust in your names, not his, usually styled as a family trust. This is useful if the property is located in a high probate state. You wouldn't have to worry about clawing the property back when he must vacate the property for nursing care or dies, but you already have that protection now by owning the house. I don't see what liability you are concerned about regarding the house. I agree that car ownership & insurance should be in your dad's name.