Did Dumbledore lie to Harry about how long he had James Potter's invisibility cloak when he was examining it? If so, why?

It's hard to give a definitive answer to this question because there is conflicting information, but here are a few points:

Note the wording of the letter: "Dumbledore’s still got his Invisibility Cloak". This indicates that Sirius was already aware that Dumbledore borrowed it, and Lily is merely telling him that Dumbledore has not returned it yet. If Dumbledore had borrowed the cloak after Harry's birthday then this letter would necessarily be at least the second communication to Sirius since the birthday. As such it would be odd that Lily is first talking about Harry's birthday in this letter, and not in the previous communication. The contents of this letter imply that it is the first communication to Sirius since Harry's birthday. This would point to the fact that Dumbledore had borrowed the cloak before Harry's birthday, which means he indeed must have had it for at least three months contrary to his statement to Harry.

On the other hand, when Harry reads the letter he seems to assume that it was written shortly before his parents' death:

Wormy was here... Pettigrew, the traitor, had seemed "down," had he? Was he aware that he was seeing James and Lily alive for the last time?

Harry, who knew that there was a three month gap between his birthday and his parents death, still seemed to think that "last weekend" could have been the last time Pettigrew would see Harry's parents, which would likely place it much closer to Halloween than to Harry's birthday.

However, it is possible that Harry didn't really fully think it through, and/or he simply made a mistake.

Another clue from the letter is the following line:

but that was probably the news about the McKinnons;

We don't know much about the McKinnons, but Moody gives us one bit of information about them in Order of the Phoenix when he shows Harry the photograph of the original order:

That’s Marlene McKinnon, she was killed two weeks after this was taken, they got her whole family.

We don't know exactly when the photograph was taken, but we can perhaps infer something from how Moody describes the rest of the dead people in the photograph:

That’s Frank and Alice Longbottom —”

and that’s Emmeline Vance, you’ve met her, and that there’s Lupin, obviously... Benjy Fenwick, he copped it too, we only ever found bits of him...

“That’s Edgar Bones... brother of Amelia Bones, they got him and his family too, he was a great wizard.. Sturgis Podmore, blimey, he looks young... Caradoc Dearborn, vanished six months after this, we never found his body... Hagrid, of course, looks exactly the same as ever... Elphias Doge, you’ve met him, I’d forgotten he used to wear that stupid hat... Gideon Prewett, it took five Death Eaters to kill him and his brother Fabian, they fought like heroes... budge along, budge along...”

“That’s Dumbledore’s brother, Aberforth, only time I ever met him, strange bloke... That’s Dorcas Meadowes, Voldemort killed her personally... Sirius, when he still had short hair... and... there you go, thought that would interest you!”

Harry’s heart turned over. His mother and father were beaming up at him, sitting on either side of a small, watery-eyed man Harry recognized at once as Wormtail: He was the one who had betrayed their whereabouts to Voldemort and so helped bring about their deaths.

Notice how Moody only gives a time frame for the McKinnons and for Caradoc Dearborn, two weeks and six months respectively. It would seem Moody was emphasizing the immediacy of those events, which might imply that the other deaths happened significantly later. If that is the case then it would make sense that James and Lily were killed several months after the McKinnons, which would be consistent with Lily's letter to Sirius being dated around the time of Harry's birthday, once again indicating that Dumbledore's statement to Harry was incorrect.

On the other hand, there is perhaps an indication that the letter was in fact written shortly before they died. Note the following statement:

James is getting a bit frustrated shut up here

The fact that James was "shut up" might imply that the Fidelius Charm had already been implemented. In Prisoner of Azkaban Cornelius Fudge describes when that was done:

"And then, barely a week after the Fidelius Charm had been performed —"

"Black betrayed them?" breathed Madam Rosmerta.

If the letter was written after the Fidelius Charm was performed and the charm was performed a week before Voldemort attacked them then the letter would have been written within a week of Halloween. This would support Dumbledore's version of the Invisibility Cloak timeline.

However, it is not necessarily clear that "shut up" would refer only to the Fidelius Charm. In this question I noted that Fudge in the very same conversation also indicated that Dumbledore advised the Potters to go into hiding immediately upon learning that Voldemort heard part of the prophecy:

According to Fudge in chapter 10 of Prisoner of Azkaban:

"Not many people are aware that the Potters knew You-Know-Who was after them. Dumbledore, who was of course working tirelessly against You-Know-Who, had a number of useful spies. One of them tipped him off, and he alerted James and Lily at once. He advised them to go into hiding. Well, of course, You-Know-Who wasn't an easy person to hide from. Dumbledore told them that their best chance was the Fidelius Charm."

So it seems that as soon as Snape told Dumbledore that Voldemort was planning on going after the Potters, Dumbledore told the Potters to hide.

As I discussed in that question, it is hard to understand why they would have hid for so long without the Fidelius Charm, but be that as it may, it is thus possible to understand "shut up" from Lily's letter as referring to the longer period of hiding, in which case it would still be consistent with the other information pointing to the letter having been written earlier.

Based on the above arguments, I would think it more likely that Dumbledore's claim of "a few days" is incorrect. However, that does not necessarily mean that Dumbledore was maliciously or deliberately misleading Harry. He may have simply made a mistake. Granted Dumbledore is supposed to be brilliant, but keep in mind that he is talking about events that occurred 16 years previously. The precise amount of time that he had the cloak for may not have been an important detail in his mind so he may not have remembered it correctly. He may have just been remembering that he borrowed the cloak shortly before they died, and without remembering the exact timeline he just stated that it was for a few days. Indeed, in his letter to Harry in Philosopher's Stone, Dumbledore simply used the vague phrase "before he died":

Your father left this in my possession before he died.

On the other hand, at the end of Philosopher's Stone Dumbledore does lie to Harry about the circumstances of the cloak, when he makes it sound like it was a coincidence that James gave it to him:

"Ah — your father happened to leave it in my possession, and I thought you might like it."

However, at that time Dumbledore was nowhere near ready to tell Harry the truth about the Hallows, so the bit of misinformation is understandable. By the time he tells Harry everything at the end of Deathly Hallows there wouldn't be much of a reason for him to deliberately lie about how long he had had the cloak for.

Of course, as noted in the question, this is very possibly simply an authorial mistake. However brilliant and truthful Dumbledore is supposed to be in the story, he is still at the mercy of the author not tripping him up by forgetting other details.