The 21-year-old striker agreed a temporary switch to Oakwell in the final throes of transfer deadline day a week ago, with both sides announcing the deal around an hour after the window had officially closed.
In a statement, Barnsley said that the transfer was announced "with the full belief" that the deal was done but "the administration process of the transfer was deemed to have not been met in order to allow the signing to be completed" upon review by the EFL.
"We are naturally extremely disappointed, but we respect the English Football League's decision and thank them for their communication throughout the process," said Gauthier Ganaye, chief exec at Barnsley.
"Oli showed a huge desire to come and play for Barnsley Football Club which can only be admired and ultimately this decision will have the most impact on him. We are disappointed, but understand that although the player was identified as a significant target for the club over two months ago, we were unable to try and effect the registration until the last moments of deadline day.
"We wish Oli nothing but the best of luck back at Swansea City in their Premier League campaign."
A spokesperson for the Swans said: "We are naturally disappointed for Oli having gone through the process, but we are pleased to have a player of his quality available again for both the first team and in a competitive under-23 structure.''
McBurnie made three substitute appearances for Paul Clement's side in the Premier League before the international break.