There are actually a few examples of this very practice, and the risks that came with it.

I'll start with the sons of Henry II of England. Henry and his wife, Eleanor, had four legitimate sons who reached adulthood. The eldest, Henry the Young King, and the third, Geoffrey, Duke of Brittany, were known to participate in Tournaments. There are also stories about Richard, but most of those are from later sources and are romanticized to the point of being total fantasy. Now, a Twelfth Century tournament was quite a bit different from how we think of tournaments now. Jousting was more of an exhibition event, rather than the main focus, and the real main event was a melee in which combatants would fight on horseback or on foot with the goal of taking each other "captive" by disarming or forcing the other to yield. This is a bit closer to how tournaments are depicted in Merlin, though Merlin tends more towards a series of one-on-one duels. The main point of a tournament during this period was to practice for the real battlefield. The social and cultural values of the medieval aristocracy of England and France at the time dictated that nobles did not often kill one another, even in regular combat, so death on the tournament grounds may not have been much of a concern (though certainly not completely ignored).

Henry the Young King participated extensively in tournaments across Northern France as a means of increasing his wealth and chivalric reputation. Matthew Strickland, biographer of Henry the Young King, discusses the Young King's tournament career at length, and its impact on chivalric culture surrounding tournaments:

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Before the Young King, kings of England had not patronized or personally participated in the tournament. Henry II permitted tournaments to be held on the borders of Normandy, but had strictly prohibited them in England, fearing they would be a source of potential disorder and rebellion. In doing so, he was re-establishing a similar ban imposed by Henry I, which had lapsed during the troubles of Stephen's reign. Henry II, like his grandfather, was a brave and seasoned warrior, but considerations of status and regal dignity, as much as concerns for personal safety, kept him aloof from tourneying himself. Henry II's reticence was shared by the kings of France. Louis VII was not a martial king, but neither Philip Augustus nor his son Louis VIII would participate in tournaments despite much campaigning in actual warfare.

Strickland then includes a quote from Ralph of Diss, a chronicler of the day:

Young King Henry, the king's son, left England and passed three years in tournaments, spending a lot of money. While he was rushing all over France he put aside the royal majesty and was transformed from a king into a knight, carrying off victory in various meetings. His popularity made him famous; the old king was happier counting up and admiring his victories, and although the Young King was still under age his father restored in full his possessions which he had taken away.

Strickland continues after the block-quote:

Far from disapproving of his son's involvement, Henry II took paternal pride in his high-profile sporting achievements and was more than willing to bankroll the Young King's tourneying activities. For he realized that the tournament provided young Henry and his companions with a vital safety valve through which to exercise their martial aggression, as well as a means for his son to enhance his status and reputation. Thus preoccupied, the Young King might be less minded to press his father for direct rule of any part of the Angevin dominions... Henry II fully recognized that the Young King's lavish patronage of an international body of knights reflected the wealth of the Angevin empire and augmented its prestige. The Young King's leadership in the tournament projected a strong a vigorous image of Plantagenet rulership. Indeed, young Henry began a new trend by which English kingship took on a self-consciously chivalric dimension, and employed royal participation in the tournament as a crucial means of forging bonds between the king and his aristocracy: after a hiatus under John and Henry III, this would be developed by Edward I and reach its apogee with the enthusiastic tourneying of Edward III and the Black Prince.

A very long quote, but Strickland's summary is very apt and straightforward. For some context, Henry II's sons, the Young King, Richard and Geoffrey, had rebelled against him in 1173-4 because he had not given them enough of their inheritance to satisfy their expectations. While Henry II did give Richard and Geoffrey more control over their respective territories of Aquitaine and Brittany, the Young King, it seems, had to make due with a larger tournament expenditure. But Henry II also saw the value of the friendships and reputation his heir would gain under the camaraderie of the tourney field. It's quite similar to how we might go to a business networking event to build a social relationship with our colleagues.

Henry the Young King had a successful tournament career but went on to die while fighting against his younger brother, Richard, in 1183. Only three years later, their brother Geoffrey, Duke of Brittany, died during a tournament in Paris. Some accounts say that he was trampled to death after falling from his horse, while other merely say that he fell ill, which has led modern historians to speculate that the tournament and/or Geoffrey's cause of death were perhaps a fabrication to cover up the fact that he may have really been in Paris to conspire with Philip II against his father or brother, Richard. This demonstrates that while the tournament grounds could be the forge for friendships, they could also give conspirators an easy way of meeting without arousing suspicion. I could not find any indication of any policy changes towards tournaments following Geoffrey's death, or anything specifically about Henry II's response, which indicates that he may really have died of an illness.

As Strickland states, this practice reaches it's zenith with Edward III and his son, Edward the Black Prince (accurately depicted in A Knight's Tale entering a tournament in disguise, though this behaviour is more commonly associated with his father). However, after the Black Prince and Edward III died, leaving the throne to Richard II, who was just a child, the practice declined. Richard II was deposed by Henry IV, whose son Henry V spent most of his time occupied with real warfare, which was passed to his infant son, Henry VI, whose reign was also filled with wars against his own family members.

We do not really see the tournament in the same light in English history until the 16th Century, when the practice was revived by Henry VIII. This Henry is a good example of a son being barred from tournament participation, as he was the only son of Henry VII after the death of his brother, Arthur. Teenage Henry was only allowed to practice with a lance by catching rings with the lance instead of charging a live enemy. A major reason for this change was that the Tudor dynasty was young; Henry VII was the first ruler and had taken the throne by right of conquest. The Plantagenets who came before them had often had scores of relatives who might continue the dynasty should an heir die unexpectedly, but Henry VII, himself an only son, had no such security. Not to mention that the very mortality of Kings had been severely shaken throughout the 15th C as the Wars of the Roses saw two rival dynasties begin systematically killing each other off. It was no wonder Henry VII became paranoid that his line should continue, which he passed on to Henry VIII. Although... one of the first things that Henry VIII did after actually becoming King was fight in a real tournament.

In summary, beginning with Henry II, it was considered a valuable prestige-builder to have your son become successful at tournaments. However, as the mortality of Kings was shaken and dynasties shrank, Kings became more protective of their sons.

I hope this answers your question somewhat, though it is quite confined to English practices. There may be more examples from other European sources of father's responses to their sons participating in tournaments. I would certainly be interest in Later Medieval French and Iberian examples if anyone would like to add to my answer.