Can a foreign national tourist "open carry" a firearm in Yellowstone National Park?

(IANAL) Technically at a Federal level I think you cannot do this unless you meet some specific requirements. From the ATF:

May a nonimmigrant alien who has been admitted to the United States under a nonimmigrant visa possess a firearm or ammunition in the United States?

An alien admitted to the United States under a nonimmigrant visa is prohibited from shipping, transporting, receiving, or possessing a firearm or ammunition unless the alien falls within one of the exceptions provided in 18 U.S.C. 922(y)(2), such as: a valid hunting license or permit, admitted for lawful hunting or sporting purposes, certain official representatives of a foreign government, or a foreign law enforcement officer of a friendly foreign government entering the United States on official law enforcement business.

From a random explanation of 18 U.S.C. 922(y)(2)

(2) Exceptions.—Subsections (d)(5)(B), (g)(5)(B), and (s)(3)(B)(v)(II) do not apply to any alien who has been lawfully admitted to the United States under a nonimmigrant visa, if that alien is—

(A) admitted to the United States for lawful hunting or sporting purposes or is in possession of a hunting license or permit lawfully issued in the United States;

(B) an official representative of a foreign government who is—

(i) accredited to the United States Government or the Government’s mission to an international organization having its headquarters in the United States; or

(ii) en route to or from another country to which that alien is accredited;

(C) an official of a foreign government or a distinguished foreign visitor who has been so designated by the Department of State; or

(D) a foreign law enforcement officer of a friendly foreign government entering the United States on official law enforcement business.

However, as per this answer to a previous Travel question about a foreigner buying ammunition leads to this ATF page:

Does the prohibition on the receipt and possession of firearms and ammunition by aliens in nonimmigrant visa status apply to nonimmigrant aliens who lawfully enter the United States without a visa?

No. A nonimmigrant alien who is lawfully admitted to the United States without a visa (e.g. Visa Waiver Program), may acquire or possess a firearm in the United States.

So it depends on how you enter the country.

But of course I would want to be carrying some pretty convincing proof of both your status, the status of your friends and the status of the weapon that you intend to borrow. At the very least you could run into Civil Forfeiture which means Law Enforcement could simply take your weapon away from you regardless of how legal it is for you to be in possession of it. (EG see Civil Asset Forfeiture: 7 Things You Should Know)

Which brings us to the individual states you mentioned:

Utah

Permit required for open carry? Hand Guns: yes, but also:

Open carry of firearm without a permit is allowed as long as the gun is at least two actions from being fired, e.g. 1) rack the slide to chamber, and 2) pull the trigger; or must carry with no bullet in the next chamber in a revolver, so have to pull the trigger twice to fire. Open carry of a loaded firearm (e.g., a live round of ammunition in the firing chamber) is allowed with a permit.

Idaho

Permit required for open carry? No

Wyoming

Permit required for open carry? No

But of course we should review what the park itself has to say on these matters: Can I Bring a Gun to Yellowstone?

Yes, you can carry a gun in Yellowstone. But it's illegal to fire it - even in self defense. And once you exit Yellowstone, you could be in one of three states, so it's important to know the law.

(My emphasis to point out a weird fact) That site also breaks down various other facts about guns and Yellowstone.

That wasn't an official National Parks Service link (I was duped). But here is the official word:

Firearms in National Parks

Possession of Firearms in National Park Units

In areas administered by the National Park Service, an individual can possess a firearm if that individual is not otherwise prohibited by law from possessing the firearm and if the possession of the firearm complies with the laws of the state where the park area is located. 54 U.S.C. 104906.

...

Use of Firearms

Unless authorized, the use or discharge of a firearm within a park area is prohibited. 36 CFR 2.4(b) and 13.30(c). In parks where hunting is specifically mandated or authorized by federal statute, firearms may be used to hunt in accordance with NPS regulations and state laws. 36 CFR 2.2.

Visitors should not consider firearms as protection from wildlife.

The really big question is if you can safely use said weapon to protect yourself?

I would suggest that you also review questions and answers on The Great Outdoors such as

Trail gun for protection against bears

What do I do if a black bear is right outside my tent?

Under what conditions can a guide be highly confident of protecting clients against brown bears without using firearms?

What to do if woke up in a tent hearing a bear/boar or something else close by?

etc