The WP article "Burning of Washington" does not give specific times, beyond vaguely stating "that night". (Confusingly, the contemporary illustrations are a mix of day and night scenes...)
However, it links to Allen (2001) History of the United States Capitol: A Chronicle of Design, Construction, and Politics, which gives exact times on page 120:
Fires in the Capitol began to be set just after nine in the evening. (...) Around 11 o'clock in the evening, the President's House was burned. The torch was put to the War and Treasury departments the next morning.
It would have been dark by this point - sunset at 39N in late August is about 18:40 local time, and astronomical twilight would be over by 20.15.
I have been poking at Stellarium and I think that at 11pm there would have been some moonlight - a half-moon, low in the SW, and setting just after midnight. But I would double-check that before relying on it!
Added: there was a question about contemporary sources. The various documents in American State Papers (Mil. Affairs., Vol 1, pp. 524-599) don't give any specific time for the burning, but has a few notes that all indicate British troops did not reach the city until around 8pm, as night was falling.
"...at twenty minutes past eight, P.M. I received incontestable proof ... that the enemy was in complete possession of the city" (Tingey, p. 577)
"at twenty minutes past eight ... the matches were applied [to the Navy Yard] ... in the way out of the branch we observed the capitol on fire" (Tingey, p. 579)
"their advance, however, reached the capitol about dark or eight o'clock" (Catlett, p. 585)
"...the enemy, who took possession of the [President's] house soon after. This now being near night..." (Simmons, p. 597)
Added2: I have tracked down a copy of Pitch (2001), The Burning of Washington, thanks to archive.org. It confirms the time given in Allen for the Capitol: the diary of John McElroy, a lay brother at Georgetown, recorded "the British ... fired the Capitol about 9.06" (Pitch, p. 106).
No explicit time is given for the White House fire, but it was after the Capitol had been burned, and there is a note that General Ross set up his lodgings at the Suter house "about an hour from midnight" (p. 116) and then went to the White House shortly before the fire was set. This is presumably where the Allen estimate of "around 11" came from.