The story goes it was the flag Bishop's father served under (hence it being WW2-era). It is according to an unfilmed scene in the original script.

I'm just gonna throw this out there, maybe it's stupid and you may have a nice laugh, but perhaps the burned flag has no significance or background story.
I'm not completely sure when this movie was filmed exactly, but it was released not long after 9/11. Again, this is just a suggestion, but in keeping with the fact that movies aren't filmed in the same sequence of events they're played out on screen, perhaps the burned flag was a reference to 9/11. Maybe they filmed it after the attacks and put it in just before it's release? Who knows.
Like I said, it might be a stupid idea, but I think, if I remember right, the volunteers who worked at Ground Zero in the months after the attacks did find a burned American flag which they hoisted up on a pole. Maybe the flag in this movie was a reference?

Good eye, I missed that. Much to my shame.

No, it couldn't. Those events happened when there were 50 states. This flag has 48 stars. I bet it comes from someplace like Pearl Harbor (maybe USS Arizona?).

It could also come from the Beirut Embassy or Marine Barracks.

You're correct Michael. It DOES have the stacked stars rather than the 50-star in-out pattern. This means the flag has 48 stars. This means it's between 1912 (admission of New Mexico, Arizona) and 1959 (admission of Alaska). I don't know of any flags that might have been partially burned that might have prestige value. As a result, I can only think that it might have some OSS/CIA related value to Muir. Almost certainly, I believe, a WWII flag Corregidor maybe? Wake?

I think the flag was older than that - didn't it have the 'stacked stars' pattern rather than modern alternate layer pattern?

That would seem to make the most sense as far as the flag coming from somewhere/thing related to Viet Nam, where he was intimately involved. Your symbolic reference also seems to add to the story. Thanks.
I almost numchucked you, you don't even realize!

I prefer to think that it's from the fall of Saigon in '75. It could have come from the airport or from the embassy. The burned part would perfectly symbolize America's "defeat" and would resonate with him, reminding him of the danger of hubris. There's some meta too with Gladys, since the CIA tried to burn HIM and his protege as well.
But no, the audience doesn't SPECIFICALLY discover what the origin of the flag was. It's up to each viewer to put it in his or her own context. But Saigon fits as well as any.

Definitely this was a misssion of significance, but does the audience ever find out SPECIFICALLY what that particular mission is?
I almost numchucked you, you don't even realize!

I thought it would have been from a past mission that he did, or something else that required him to be in a dangerous situation and he was able to come out with it as a 'souvenir" from that event.