G F Black's The Surnames of Scotland doesn't suggest an origin for the name.
However the earliest reference in Scotland is in 1492, when Martin Divorty was vicar of Petty (a few miles east of Inverness). If Wikipedia is correct in attributing the term Huguenot to the Genevan burgomaster Besançon Hugues (1491–1532), then this Divorty pre-dates the Huguenots by some decades.
Black also offers a large range of variants, including Divertie, Dovertie, Durty, Dirty, Durtty, Dwertty, Douerty, Devertie, Dovartie, Dovertie, Davortie, Divertye, Doverty, Dortye and possibly Doraty.
A Quick Search on SP for D*v*r*t* finds 5290 results, though some are unrelated names (e.g. Davenport) and there is at least one Durvart who is more likely to be a variant of Dorward. And of course that search doesn't find variants where the letter v is omitted.