Paul Burrell has sued Prince Harry for lying. In fact, there is no news story that reports that Burrell has even threatened to sue Harry.
The claim that Burrell is suing Harry for lying is a rumor that has been circulating online. There is no evidence to support this claim, and it has been widely debunked by fact-checkers.
Burrell was a former butler to Princess Diana, and he wrote a book about his time working for her in 2003.
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In the book, Burrell made some claims about Diana's private life that were not well-received by the royal family. Harry was particularly critical of Burrell, and he accused him of lying and profiting from Diana's death.
However, there is no evidence that Burrell has ever sued Harry for lying. The rumor that he is doing so is likely based on Harry's past criticism of Burrell. It is also possible that the rumor is being spread by people who are trying to damage Harry's reputation.
Until there is evidence to the contrary, it is safe to say that Burrell has not sued Harry for lying.