The Mongol Empire was pretty fragmented, but various of its branches had large effects on the Byzantines.

Genghis Khan himself made it to the Dnieper, and Subutai destroyed Moscow and Kiev in the mid-13th century, throwing a large source of trade (down the Dnieper, through the Black Sea, into the Mediterranean by way of Constantinople) into complete disarray for the Byzantines. Of course, at this point, the Byzantines had been kicked out of Constantinople by the Crusaders, and trade was withering and dying anyway.

South of the Black Sea, Hulagu Khan captured Baghdad and deposed the last Abbasid caliph in 1258. This brings up an interesting point, however. For most of their later history, the Byzantines have been shown as completely opposed to (or at least antagonistic to) all Turks/Turkmen/Central Asians. But the fragmentary nature of the Mongol Empire and its successors often gave rise to situations in which Central Asian armies were among the Byzantines' best allies. And confusingly enough, it was altogether common for one group of Central Asians to be fighting against the Byzantines while another group of Central Asians was fighting against the first group of Central Asians.

The best example of this is during the reign of the Ottoman Turkish sultan Bayezid I. Bayezid was a major enemy of the Byzantines, laying siege to the capital for nearly ten years, ending in 1402. I think it likely that such constant pressure would have eventually caused the city and the empire to fall, but the siege was actually broken by the entrance into Asia Minor of Timur, another Turko-Mongol leader, whose armies swept away major cities controlled by the Ottomans. Bayezid was eventually defeated by the Timurids, and Constantinople survived another 50 years.