To "give the lie" is an English expression meaning to expose a lie, or show a thing is not true. It is still in use today.
to show that something is not at all true
These figures give the lie to the notion that people are spending less.
to prove that something is not true:
The fact that the number of deaths from cancer in the area has doubled surely gives the lie to official assurances of the safety of nuclear power.
So the literal interpretation of Macduff's words is that drink made the Porter tell lies. This being Shakespeare though, there are deeper meanings at play.
The most obvious one is a pun: drink made the porter "lie" as in "lie down drunk". Drink also makes people into "liars" because they talk nonsense.
A more interesting interpretation, however, is that the Porter in this scene can be read as a stand-in for his master, Macbeth. His speech about hell is a metaphor for the hell which Macbeth's castle has become and each of the three sinners for an aspect of Macbeth's character.
In this interpretation, Macbeth has been "given the lie" by the Witches through their manipulative prophecies. The result is his confusion, as though drunk with power and aggression, and eventually the literal "lie down" of his death.
While "give the lie" remains a current phrase, the Porter's reply is another figure of speech which has become less common:
lie in one's throat
to tell a foul or outrageous lie
So he is simply confirming that Macduff is correct via a comic overstatement. This then adds dramatic weight to his "requitation" of the drink, managing to stand and converse despite his hangover.
It's worth noting that again there is a literal pun here. When drinking the drink does of course "lie" in the throat of the drinker both in drinking and in the aftermath of alcohol-provoked vomit.
Macbeth and His Porter, Frederic B. Tromly, Shakespeare Quarterly, Vol. 26, No. 2 (Spring, 1975), pp. 151-156