Gar is a northern English and Scottish word that can mean to make, to cause. The phrase "gars me pine" means roughly "makes me feel sad". The OED has many citations showing different shades of meaning, few from the last century or so. If you google "gars me pine" you will find it in places like the 1885 poetry of Alick Murray whose poem "Oh! Cauld Cauld" says
It's no th' cauld that gars me pine,
Nor nature's looks sae bleak an' drear ;
But as th' earth lo'es warm sunshine
To me Love's smiles are sunlicht dear!
At times I've seen a gowden chink
Break thro' th' veil o' stormy clouds,
An' hope assures a bonnie blink
Will rend fell anger's frowning shrouds!
(That is, his sweetheart's cold looks make him feel sad, but sometimes she gives a hint that she does like him.) This was apparently published the same year Burton published his Arabian Nights, so the phrase was not completely unknown or incomprehensible to Burton and his audience.