The Gemara brings a prohibition of shaking off a garment on Shabbat. Rashi understands this to mean that one who shakes dust off a garment violates the Shabbat, and is liable to bring a Korban Hatat. However, others learn that the prohibition applies only to shaking dew or water off a garment, but not dust. Shulhan Aruch clearly did not learn like Rashi, but like the other opinions. He does not record any prohibition regarding dusting off a garment. The Rema does cite the opinion that it is prohibited to do so to a garment that he is particular about and adds that one should follow this position. The Ben Ish Hai follows the Rema and prohibits dusting off a garment. Hacham Ovadia and Hacham Bension were lenient, but they agreed that one who is stringent is praiseworthy and that it is preferable to do so with a Shinui (unusual method).
The Mishna Berura (302:2) cites the Elyah Rabbah that dusting a garment is prohibited by law. He bases his ruling on the many other authorities that hold like Rashi. However, even these strict opinions allow one to instruct a non-Jew to dust off a garment, especially since it is often an issue of Kavod HaBriyot (human dignity) to have presentable clothing. The Mishna Berura concludes by advising that one avoid getting into the debacle in the first place by taking care not to leave his clothes or hat in a dusty place.
SUMMARY
According to Maran, it is permitted to dust off a garment on Shabbat. However, one should ideally avoid doing so and take care not to leave his garments in a dusty place. It is certainly permitted to instruct a non-Jew to shake off the dust.
