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Well, first of all, it does a little bit, though more often his conscience bothers him BECAUSE it isn't bothering him over lies. He knows he should feel bad but doesn't, and that makes him feel bad. But in a larger sense, it's an indication of the contradictions within Huck. He's been taught to fear hell and to live a moral life, and he wants to please those forces of society, as epitomized by the Widow. And yet he enjoys his adeptness at falsehood, which confuses him. It's a similar confusion to how he is supposed to feel about slavery and how he actually does. The point is that some morality is instilled/forced into us, while some morality we find ourselves.
Huck is a survivor. He lies as part of his instinct as an orphan. Part of Huck's boyish charms are his lies. They are often colourful and imaginative. Most of Huck's lies are not done out of malice rather than playfulness and opportunity. Lying does bother Huck when people get really hurt. Huck feels badly when he finds himself part of the King and Duke's plan to defraud a family out of their inheritance. Huck even attempts to foil their plan.