On the whole, no. You will have been given the task because the king believed you were rich and powerful enough to accomplish the task and that you had the resources to do it.

Where you got those resources would probably depend on who you are and what kind of task you were undertaking. Delivering a message is quite vague, so lets take two examples. Firstly, say the King is in Paris and needs a letter delivered to the archbishop at Reims. It would probably be taken by one of his servants or retainers, a high-level member of his household. As a member of the King's retinue, you would probably be drawn from the nobility (maybe a younger son or an heir), probably from the Paris region (the Ill-de-France). The king would feed you and put a roof over your head, provide you with gifts and materials suited to your station (such as a horse, weapons or clothes) - in addition to what you owned already. This wasn't a salary, as such, but would be rather ad hoc, a way of securing your service. If you were particular high in his favour, you might be granted some lands or even a lordship, a castle of your own. In return for this, you would also be expected to do whatever the king asked, like take a letter to the archbishop of Reims, fight in any campaigns the king was running, attend his court, advise him, etc. etc.

Say the message is a bit more important an embassy to the pope, for example. This would likewise go to someone more powerful and more influential - in this case, probably to an important cleric, maybe the bishop of Orléans or the abbot of St-Denis. As someone in this position, you would have a considerable amount of institutional wealth at your control and would rely on the king as your prime protector and patron (after all, it was he who confirmed you in office). You would use your own personal wealth (which came in most part from the lands associated with your office) to complete the mission, pay for food, horses, guards etc.

The biggest jobs, administering chunks of the country, went to the most powerful people, the higher aristocracy, the counts and the dukes. Here, the king really had very little power to influence things, except in exceptional circumstances. The counts and dukes usually ruled their territories because they were the most powerful and richest people in them, and since there was no salary or expense provided by the crown, any attempt to impose another claimant would be doomed to failure. Only in extraordinary cases could the king impose a candidate on a territory, and usually then it required a death without clear heirs and usually some fighting against a rival claimant: a good example can be seen in King Robert II's Burgundian wars in the 1000s and 1010s, when trying to put his son Henry (the future Henry I) into the office of Duke of Burgundy.

This answer works for France for the 12th century. Things do start to change, veeeeery slowly in the 13th century, as the French monarchy becomes much more powerful with the conquests of Philip II and more centralised, particularly in the reign of Louis IX and Philip III. I know much less about this period though, although I still don't think there was anything close to what we would recognise today as a system of expenses.