The Byzantines actually sought and gained a moderately successful alliance with the Ilkhanate and the Golden Horde. http://en.wikipedia.org/wiki/Byzantine%E2%80%93Mongol_alliance

The wikipedia article doesn't elaborate but the conversion of the Khanates to Islam was a major factor in the breakdown of relations. Prior to that the Byzantines thought they would make useful allies against the Muslim dyanasties of the region.

Perhaps the greatest impact of the Mongols on Byzantine territory, however, ended up being the sort of cueball effect as Turkic military bands streamed into Anatolia as the Turkic sultanates crumbled in the Mongol invasion.

The most famous of these Turkic tribes, the Ottomans, would of course conquer Byzantium, but there were numerous Ghazi states in the Byzantine march land over the centuries. http://en.wikipedia.org/wiki/Anatolian_beyliks

The fall of Constantinople to the Fourth Crusade in 1204 coincidentally coincides with the official founding of the Mongol Empire in 1206. After this initial fall of the Byzantine Empire, control of the Anatolian region was divided between Greek successor states and the Seljuk Turks. The Seljuqs were themselves part of the many Turkic east-to-west migrations throughout history, and had been in multiple wars against the Byzantines. After the Battle of Köse Dağ in 1243, the Mongols defeated the Sultanate of Rum (a Seljuk state), making them vassals together with the Empire of Trebizond, a Byzantine successor state.

After Michael VIII Palaiologos restored the Byzantine Empire and recaptured Constantinople in 1261, he sought an alliance with the Mongols, by then already divided into the Golden Horde and the Ilkhanate. He signed a treaty in 1263 with the Mongols and married off two of his illegitimate daughters to Nogai Khan of the Golden Horde, and Abaqa Khan of the Ilkhanate. After Abaqa died, his wife Maria Palaiologina went to Constantinople to become a leader of a monastery now known as the Church of St. Mary of the Mongols, named in honor of Maria (the "of the Mongols" part).

Michael VIII's successor, Andronikos II, also sought an alliance with Ghazan Ilkhan. Wikipedia states that " The death of Ghazan in 1308 was mourned by the Byzantines". Relations continued under Oljeitu. He also tried to create an marital alliance with Toqta and later Ozbeg of the Golden Horde. Relations seem to have turned sour after this time period, with the Mongols also turning toward Islam as a state religion.

After Andronikos II's succesor Andronikos III, the Byzantine Empire began to decline, with the Ilkhanate also disintegrating by 1335. The region is eventually conquered by the Ottomans.

This Wikipedia article on the Byzantine-Mongol alliance is a good read