Are there any specific events which grants a soldier permission to execute a prisoner on sight? (Like in the movie: Fury)

The Hague Convention of 1907 on the rules of land warfare, ratified by the US a few years later, prohibits the following (Section II - Hostilities):

"To declare that no quarter will be given" (quarter is the military term for mercy).

On the treatment of prisoners of war, the following is said:

"Prisoners of war shall be subject to the laws, regulations, and orders in force in the army of the State in whose power they are. Any act of insubordination justifies the adoption towards them of such measures of severity as may be considered necessary." This implicitly refers to the Lieber code.

Compare this to the administrative document mentioned by Pieter Geerkens, the Lieber code. If the Lieber code mentions "no quarter will be given under such and such circumstances", then this section should not be binding.

Source: http://avalon.law.yale.edu/20th_century/hague04.asp

About shooting a German soldier because he is wearing a US uniform coat. In the American Civil War many rebel soldiers acquired US uniform coats, jackets, etc. from dead or captured US soldiers, and dyed them gray or butternut if they had time. And lots of those rebels were later captured without many illegal executions of those wearing clothing of Union origin.

This reminds me of the movie Indian Uprising (1952). In an opening scene (which was missing the next time I saw the movie) the troops of captain McCloud (George Montgomery) capture the teenage son of Geronimo. The soldiers are very angry and want to lynch him because he is wearing a bugler's jacket, presumably from a dead bugler boy. But the captain won't let them harm Geronimo's son.

And that reminds me of history. The 19th century Zulus were fierce and merciless warriors. Their religious rituals after killing an enemy included slashing his or her belly and taking a piece of clothing to wear for a time. After the bloody Zulu victory at Isandhlwana on 22 January 1879 rumors true or false were common among the British soldiers that the corpses of the boy soldiers killed there (Thomas J. Harrington, Robert Richards, Joseph S. McEwan, Daniel Gordon, and James Gurney) were mutilated much worse than other soldiers and/or that they had been tortured to death.

I read a description of the British fort at Rorke's Drift sometime afterward that mentioned that they had a Zulu prisoner accused of spying. I thought that prisoner had a very slight probability of escaping hanging. And then i read that the prisoner was suspected of torturing the poor little drummer boys. I wondered what evidence the soldiers could have had to suspect him beside wishful thinking. But today I realized that if (repeat if) he had been caught wearing a boy-sized piece of British uniform that would connect him to the death of at least one boy.

Anyway, it sounds like both of the killings in the movie Fury would count as illegal killings and thus murders. To be sure, consult online versions of the Geneva and Hague conventions prior to WWII and the us military code of justice in force at the time.

I have met a old man who was a Serbian commie partisan.

He said that their rule was:

• captured a Heer officer - check his papers, then hang or send to POW camp.

(It means that only known war criminals were executed. The villagers would tell them all about who the bad guys were).

• captured an SS officer - hang first, then check his papers.

Then leave the body hanging from some tree over a road used by the Germans.

In general no, though specific exceptions have existed, and continue to exist, in the Law of War code adopted by the U.S. Army.

Through much of its history the U.S. Army has been regulated by the Instructions for the Government of Armies of the United States in the Field (Lieber Code). 24 April 1863.. The current U.S. Army Rules War still exhibit much of the character of the Lieber Code. Some articles of that code are particularly relevant to the question:

Art. 12. Whenever feasible, Martial Law is carried out in cases of individual offenders by Military Courts; but sentences of death shall be executed only with the approval of the chief executive, provided the urgency of the case does not require a speedier execution, and then only with the approval of the chief commander.

Thus, a soldier of the U.S. Army is not generally entitled to carry out sentences of death for violations of the Laws of War. Rather he may only do so when his unit, out of military necessity, has been specifically delegated that authority by the chief commander. I read into the text that such authority should only be deemed to apply for a single operation at a time.

Art. 56. A prisoner of war is subject to no punishment for being a public enemy, nor is any revenge wreaked upon him by the intentional infliction of any suffering, or disgrace, by cruel imprisonment, want of food, by mutilation, death, or any other barbarity.

Art. 59. A prisoner of war remains answerable for his crimes committed against the captor's army or people, committed before he was captured, and for which he has not been punished by his own authorities. All prisoners of war are liable to the infliction of retaliatory measures.

Art. 60. It is against the usage of modern war to resolve, in hatred and revenge, to give no quarter. No body of troops has the right to declare that it will not give, and therefore will not expect, quarter; but a commander is permitted to direct his troops to give no quarter, in great straits, when his own salvation makes it impossible to cumber himself with prisoners.

Art. 62. All troops of the enemy known or discovered to give no quarter in general, or to any portion of the army, receive none.

Under this it is acceptable to refuse to accept the surrender of enemy combatants known to themselves refuse quarter; but not to exact any punishment on them after they have been granted quarter.

Art. 63. Troops who fight in the uniform of their enemies, without any plain, striking, and uniform mark of distinction of their own, can expect no quarter.

Art. 64. If American troops capture a train containing uniforms of the enemy, and the commander considers it advisable to distribute them for use among his men, some striking mark or sign must be adopted to distinguish the American soldier from the enemy.

Art. 66. Quarter having been given to an enemy by American troops, under a misapprehension of his true character, he may, nevertheless, be ordered to suffer death if, within three days after the battle, it be discovered that he belongs to a corps which gives no quarter.

Art. 81. Partisans are soldiers armed and wearing the uniform of their army, but belonging to a corps which acts detached from the main body for the purpose of making in roads into the territory occupied by the enemy. If captured, they are entitled to all the privileges of the prisoner of war.

Art. 82. Men, or squads of men, who commit hostilities, whether by fighting, or inroads for destruction or plunder, or by raids of any kind, without commission, without being part and portion of the organized hostile army, and without sharing continuously in the war, but who do so with intermitting returns to their homes and avocations, or with the occasional assumption of the semblance of peaceful pursuits, divesting themselves of the character or appearance of soldiers -- such men, or squads of men, are not public enemies, and, therefore, if captured, are not entitled to the privileges of prisoners of war, but shall be treated summarily as highway robbers or pirates.

The distinction is here made between acceptable and unacceptable clandestine acts of sabotage: namely that to be regarded as a combatant individuals must be wearing the colours and insignia of their command.

In terms of antecedents, the Lieber Code can be regarded as a codification of standards already implicitly accepted, "amongst the European nations and their descendants,", since the Peace of Westphalia in 1648. In the Thirty Years War which those treaties concluded one readily finds wanton disregard for virtually all aspects of the Lieber Code; and in the years from 1648 to 1863 only occasional disregard for its principles.

Modern Geneva Conventions are generally much more conservative in their interpretation of military necessity, while the U.S. Army tends towards a modern outlook intermediate between the Lieber Code and the most modern international conventions.

Note also that the Lieber Code applies only to land warfare. Ruses generally accepted since the dawn of time as perfectly acceptable for naval warfare (flying enemy colours for example) are often grounds for summary execution under the rules for land warfare.

See also my answer to the question:
What was the custom/consensus regarding irregular combatants 19th century?.

Update: Scan of original publication, which may be more useful for some research.