If I created an irrevocable trust and left everything to my cat and specified no one else may inherit the estate, what will happen to it?

You can leave your wealth as a trust, which is not "your estate". The trust doesn't die when you do, which is the whole point for many people.  There is no probate of a trust, so "inheritance" is irrelevant.  Nobody can "challenge" the trust, to the extent you're not legally incompetent when you prepare it.

As for answers about the rest of the estate "going to the state" when the cat dies, that's unlikely.  Anyone drafting a trust would usually include dissolution provisions to AVOID that, such as simply donating the  remaining value of the trust to a local charity focused on "prevention of cruelty to animals", or the like. 

The trustee would use it to care for the cat.  

it will go to a trustee and anything will go to the care taker of the cat for the care of the cat.

Morningfox is DEAD WRONG. You would actually leave everything to the TRUST, with your cat as beneficiary. You WOULD NOT leave ANYTHING to the trustee, who only administers the trust. They would be legally entitled to a management fee paid from the trust. If you fail to name a trustee, the court will appoint one. That shouldn't be an issue, because the attorney drafting the documents creating the trust will insist you name someone as trustee, and volunteer to  be named if you have no one else.

It will go to the cat's care. People have done stranger things than that. If no provision is made for assets after the cat passes, it will likely be contested by the next of kin or divided amongst them according to intestate laws; as if there was no will. 

If you literally left it all to your cat, the trust would be invalid. You could leave it to a trustee, on the condition that she care for the cat in luxury as long as the cat lived. Then you have to specify who gets the remainder after that (else it will be your heirs).  You could specify a charity.

Then the remaindermen and the trustee will fight constantly about the amount spent on the cat. Lawyers and courts will get involved. They will argue until the legal fees eat up the whole trust, and the cat gets kicked out onto the street.

If you *really* wanted to care for the cat, you would set up the trust so that a *reasonable* allowance went to the carekeeper. Make it small enough so that it’s not worth fighting over in court.

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Maybe Steven F lives in state where you can leave things to a trust.

But in most states, the appropriate language is something roughly like this: "I leave all of my property and estate to the trustee of My Silly Cat Trust dated 31 Feb 2021, to be held and disposed of in accordance with the terms, covenants and conditions of that trust."

Relatives would go to court to contest the will, so you would have to write an affidavit of disinheritance to go along with the will, naming each likely claimant and why he or she should not receive anything.

Then you need someone you trust to take care of the cat. The cat would not know what to do with the money, so would need a human to sort it out. That person might be tempted to keep a lot of the money, because the cat is not likely to complain if the funds go missing.��

A cat can't legally own property.  If no humans, corporations, or charities can inherit, then I guess everything would have to stay in the trust.

Someone has to administer the estate so the cat could be cared for, they will charge a fee.

The trust would be ruled invalid and your estate would pass according to the laws of intestacy.