The 27-year-old, who joined Bath from Leicester Tigers in the summer, was suspended by a Rugby Football Union disciplinary panel after he pleaded guilty to the offence on the Worcester man which saw him sent off in the 63rd minute of the side's 29-13 Premiership win.
It means Burns will miss Bath's opening Champions Cup game at home to Italian side Treviso on Saturday and the trip to Scarlets on the following Friday, October 20.
The final game of his ban will be the Premiership derby with local rivals and his former side Gloucester at the Rec on October 29.
Chairman of the panel Gareth Graham said: "The panel heard evidence from the player as to the tackle and concluded that this was a reckless tackle that had, by the player's own admission, 'gone wrong' as opposed to an intentional act of foul play.
"Given that there was no injury to the Worcester player, the panel found it was a low-end entry point.
"The player accepted the charge in full, apologized immediately on-field and to the Worcester player post-match and has a clear disciplinary record. Therefore the panel gave full mitigation reducing the sanction to three weeks."