No, Peyton Farquhar did not actually burn down the Owl Creek bridge. He was, essentially, set up by the Union army so that they were anticipating his attempt to do so, and he was—we can assume—caught in the act of trying to burn it.
In part 2 of the story, we learn that, at some point in the near past, a "gray-clad soldier," who looked to be a soldier in the Confederate army, approached the Farquhars' plantation and shared some news. He said that the Union army had reached the Owl Creek bridge and that the commandant had issued an order that made it clear how very important the railroads are to the Union war effort in the South by publishing the heavy penalty to be paid by anyone who "interfer[ed] with the railroad, its bridges," and so on. The soldier also tells Farquhar that only one sentinel guards the bridge and that a great deal of dry wood is heaped up near the structure, wood that would burn quickly if set on fire. The final line of part 2 informs us that this soldier is actually a "Federal scout"—a member of the Union army—and not a Confederate soldier at all.
Though Farquhar's attempt to burn the bridge is not depicted in the story, we can assume that is what he attempted to do, lured by the "opportunity for distinction" in doing a great service to the South. Otherwise, the information in part 2 would be largely irrelevant. Further, the final line of the story makes it clear that the bridge from which he is hanged is, in fact, the Owl Creek bridge. So, we know that it is still standing, that any attempt he made to burn it was unsuccessful, and that he was, apparently, set up to fail.