I don't know if there is a current monarch whose 'heir' is not actually his, but DNA from the remains of king Richard III of Britain shows that somewhere in the line, someone cheated. See this news article

EDIT: King Richard III of England, not Britain. My mistake.

Not offspring as such, but recent genetic studies cast doubt into the paternity of Napoleon III, nephew of Napoleon Bonaparte, Emperor of the French. Full disclosure: I'm a Bonapartist.

Napoleon III's father was Louis Bonaparte, Napoleon I's brother and King of the Netherlands. He had been married to Hortense de Beauharnais, Napoleon's step-daughter (Josephine's daughter from her first marriage), and neither was happy with the match - Louis was jealous, cold, and Hortense was prone to depressive episodes. The couple often spent large swathes of time apart from each other.

Hortense was known for her infidelities, as a result. At the probable time of their second son's conception (Napoleon III had an elder brother, Napoleon-Louis who died fighting with revolutionaries in Italy in the Carbonari rebellions), Hortense and Louis were in different countries - except for one day, which supporters of Napoleon III claim was the day of his conception.

Nevertheless, his life and his reign were still marked by rumor concerning his true parentage.

Genetic studies performed to codify Napoleon I's genetic heritage showed that the descendents of his brother Jerome (King of Westphalia and ancestor to the current Bonaparte line) and the descendents of his illegitimate son through his mistress the Countess Walewska bear his patrilineal genetic marker.

However, Napoleon III did NOT share this patrilineal lineage, meaning he was not descended from Napoleon I's father Carlo, and consequently not from Louis Bonaparte.

Possible candidates for Napoleon III's true father include Charles Joseph, comte de Flahaut (the illegitimate son of Talleyrand, one of Napoleon's chief statesmen and rivals), who fathered one of Hortense's bastards, the Duke of Morny (a prominent Second Empire politician and orchestrator of the coup that brought Napoleon III into power).

To note however is that under French law at the time of Napoleon III's birth, the husband of the wife would be the legal father of the child, regardless of true parentage, which in any case could not be proven at the time.

Another possibility is that it is in fact Louis Bonaparte who was the illegitimate one, through his mother Letizia Bonaparte, and Napoleon III his trueborn son.

Sources: "Reconstruction of the Lineage Y Chromosome Haplotype of Napoleon the First," Gerard Lucotte et al. http://www.ijsciences.com/pub/pdf/V220130935.pdf

"La mort de Napoléon, Mythes, légendes et mystères," Thierry Lentz and Jacques Macé (2009)

I wrote this a while ago, but William the Conqueror may fit the bill. Not from adultery, however.

In 911 Rollo signed the Treaty of Saint-Clair-sur-Epte with King Charles III of France which allowed Normans to settle in Neustria. This was after invading Paris, but it granted the lands we will eventually call Normandy. At the time, it was known as Rouen. Rollo was given the title Count of Rouen. His successor, William I, expanded Rouen. However, many consider him a Duke of Normandy despite the title not technically existing until the 11th century.

William I (Longsword) attacked Arnulf I, Count of Flanders in 939. He, along with with Louis IV, King of France, retaliated against William I where he pledged loyalty to the King of France. He was killed in an ambush in 942.

That same year, Louis IV seized Rouen and took over the lands. Richard I, son of William I, was only 10 years old at the time. He was taken under the care by Count of Ponthie, but managed to ally himself with the Norman and Viking leaders. By 947 he claimed Rouen back under his rule.

Richard II (Richard The Good) took over rule in 996 when his father died. He married his sister (Emma of Normandy) to King Æthelred of England (important). He helped Robert II of France fight off the Dutchy of Burgandy, and later forged an alliance with Brittany with the marriage to Judith of Brittany.

Richard III took over after the passing of Richard II, but it lasted a short year (1026 - 1027). His younger brother, Robert I, was next in line. This guy alone is an interesting story in himself as it's believed he was responsible for his brother's death. Earlier, he attempted a civil war against his brother. He also assembled an army against his Uncle, caused disturbances in the civil war in Flanders, etc. He made his illegitimate son, William, heir while he went to Jerusalem. He died in 1035 probably from illness.

Remember Judith of Brittany? Yeah, her Grandson is William The Conqueror. Remember her husband? And Emma, mentioned earlier, married King of England (Æthelred) and was driven out of England by Denmark and went to Normandy for refuge in 1013. This is where William was born in 1028 by Judith's daughter Herleva and Robert the I. This is part of purpose behind William's conquest of England.

The Archbishop Robert and King Henry I of France mentored William. When the Archbishop died, Normandy went into anarchy. In 1046, with the help of Henry I, William led a rebellion in lower Normandy and eventually reclaimed all of the land back. The rest is history.

During this whole time, however, the people in Couen had always considered themselves as Norman. Even though it was fiefdom of France, it was essentially considered a vassal where Normans could reside. The term Vassal wasn't used until about 1057. I think this is where the confusion is. I think if you could relate it to modern times, it would be like Puerto Rico to America. It is technically a part of America, but they're still their own territory and government, but America allies with them to help when necessary. They don't consider themselves American, rather Puerto Rican.

One thing I didn't cover is when Normandy first started getting called Normandy, and that was about the time Richard II and is considered the first Duchy of Normandy.

Sources:

Eleanor Searle, Predatory Kinship and the Creation of Norman Power, 840–1066 (University of California Press, Berkeley, 1988)

Bates, David R. William the conqueror. The History Press, 2004.

François Neveux, A Brief History of The Normans (Constable and Robinson, 2008)

I've always wondered how this worked with Catherine the Great of Russia, because her husband, Czar Peter III, was said to be unable to produce any children with her, so she turned to other men (some allowed by the reigning Empress Elizabeth at the time, some not), to ensure an heir. This is from AlexanderPalace.org:

He was also unable to consummate his marriage to Catherine. Elizabeth didn't understand the fault was Peter's and pressured the couple to produce a son - thus securing the dynasty. When it was clear this wasn't going to happen, Elizabeth permitted an affair between Catherine and a handsome Russian officer, Serge Saltykov. Catherine conceived and bore a son, Paul, who was accepted by Peter as his own.

Two books I read recently support the idea that Sergei Saltykov was really Paul's father (Robert K. Massie's Catherine The Great: Portrait of a Woman (2011), and Henri Troyat's Catherine The Great(1978)) I've always wondered how this worked when many people believed Saltykov was really Paul's father. So how could Paul be seen as a legitimate leader if his mother wasn't even Russian and his biological father wasn't a czar?

King Christian VII of Denmark's wife, Caroline Matilde had an affair with the king's personal physician, Johann Friedrich Streunsee, and it's commonly agreed upon by many experts that the queen's second child, a Princess Louise Augusta was almost certainly Streunsee's biological child, due to a) the fact that Streunsee and Caroline were confirmed to be lovers at the time Louise Augusta was conceived, b) the King and Queen hadn't been intimate in several years prior to the pregnancy, and c) facial analysis of portraits of father and daughter show several common features.

King Christian absolutely knew about the affair and that Louise Augusta was not his child, but it had no affect on her position, she remained until the end of her days a monarch. Christian held his physician (who was executed) in very high regard even after the discovery of the affair.

A cinematic adaptation of the events can be found on Netflix, it's called A Royal Affair.

isn't this a "throughout history" question ? I'm not complaining but other questions that are interesting get bashed or removed due to that...