While speaking in the Ancient Language in the Inheritance Cycle one can't tell a lie. The closest one can come to doing so is speaking a half-truth. Thereby circumventing this rule.
But why can't someone tell a lie in the Ancient Language?
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You will find an explanation in Eldest. Oromis explains why magic is bound by the ancient language:
... afterward, a race called the Grey Folk—not elves, for we were young then—gathered their resources and wrought an enchantment, perhaps the greatest that was or ever shall be. Together the Grey Folk changed the nature of magic itself. They made it so that their language, the ancient language, could control what a spell does ... And they gave the ancient language its two unique traits, the ability to prevent those who speak it from lying and the ability to describe the true nature of things. How they did this remains a mystery ...
There is a passage when Eragon tries to say a lie (speaking to Arya) and he can't - the words cannot be said if the person thinks they are a lie
“I . . .” He tried to lie—not wanting her to know how much he had missed her—but the ancient language stopped the words dead in his mouth and rendered him mute.
Also, about Eragon's poem:
“However, I am surprised that you can give voice to it in this tongue. No barrier exists to writing fiction in the ancient language. The difficulty arises when one attempts to speak it, for that would require you to tell untruths, which the magic will not allow.”
Minor note, the truth in this context is what the person believes to be true. For example, (SPOILER WARNING)
Murtagh tells Eragon that he (Eragon) is son of Morzan. He tells it in the ancient language and he believes it to be true. We learn the real truth in Brisingr.
In the end of the 4th book Galbatorix shows that like every type of magic this spell can be broken if you know the right word.
Because the words have a power over everything, I would presume that, in order to tell a lie, you would be saying something that would make it be true. By virtue of the fact that the Ancient Language affects everything, everything you say has to be true, because it will become true when you say it. I will add some evidence from the books once I can find it.
Brom cocked an eyebrow and said, “Fethrblaka, eka weohnata néiat haina ono. Blaka eom iet lam.” A bird suddenly flitted from a branch and landed on his hand. It trilled lightly and looked at them with beady eyes. After a moment he said, “Eitha,” and it fluttered away.
Brom uses this example to show that even the animals understand the Ancient Language somewhat. Unfortunately I'm still struggling to come up with something more concrete from the text. This may just be how I understood it and complete speculation, but I thought there was more actual text.