Cases of firstborn (in the sense that they are the first children of the monarch) twins are rare and there are only a few instances of it occurring throughout history. However, they are certainly not non-existent.
A good example would be the case of the former Shah of Iran, Mohammed Reza Pahlavi who had a twin sister, Princess Ashraf Pahlavi. In this case, there was really no question of Ashraf ascending the throne, since she was born five hours after her twin brother. More importantly, women were not allowed to ascend the throne of Iran due to agnatic primogeniture which excluded females and female-line descendants, so it fell to her brother to be the heir.
A more interesting (albeit recent) example is the case of the current Hereditary Prince of Monaco, who was born minutes after his sister. However, since Monaco practiced male-preference primogeniture, meaning that males are preferred to inherit the throne than females, it was the younger male twin who was declared the heir and not the older sister.
To sum up, the answer to this question greatly varies on the type of monarchy to which the twins are born. In current monarchies like the Netherlands and Norway, which practice absolute primogeniture (which means that the eldest offspring, regardless of sex will inherit) then it would be the first child out of the womb. However, if a monarchy practices male preference primogeniture (meaning that the eldest son will inherit, regardless of the existence of elder sisters) or adheres to Salic Law (meaning that females are absolutely barred from inheriting the throne, as in the case of the defunct French royalty) then the sex of the twins/triplets will be controlling. If all of them are the same sex, then the eldest will be proclaimed the heir.
The reason we have to resort to conjecture and hypothetical situations is that situations of twins or triplets rarely arise in royal families, and less so with them being the first born offsprings of the monarch. The few examples I have mentioned above are from recent history.
There have actually been a few examples of twins in royal families over the centuries. One example was the fraternal twins Louise Elizabeth and Anne Henriette of France (born 1727). They were the eldest children of Louis XV of France. The firstborn son, also Louis, was born in 1729 and, as mentioned by another commenter, the French practiced Salic Law, there was never any question that Louis was the heir. There are several other similar examples of female fraternal twins.
However, more pertinent to your question would be the case of the twins Berenguer Ramon II and Ramon Berenguer II of Barcelona (born either in 1053 or 1054). These two were, in accordance with their father’s wishes, to co-rule the County of Barcelona after his death in 1076. This they did for a time, which seems to have been characterised by an inability to agree on just about anything. Eventually, they simply divided the County between them. Ramon Berenguer died in a hunting accident in 1082 after which his twin became sole ruler. However, the widespread belief that Berenguer Ramon had orchestrated his brother’s murder led to civil war and an eventual compromise whereby he would rule until his brother’s son, Ramon Berenguer III came of age in 1097. However, this is something a special case since it was their father who explicitly wanted his twin sons to co-rule, rather than a case of following any sort of rules of succession, though it seems likely that their father wanted them to co-rule precisely because he was unable to apply normal rules of succession and choose which should be his heir. Perhaps he did not know for sure which was actually the firstborn.
Another case was James II of Scotland, who was the younger twin brother of Alexander. Alexander stood to inherit the throne (as evidenced by the granting of the title Duke of Rothesay which was given only to the Scottish heir apparent) but died in infancy.