The fourth crusade had some impact on the beginning of the Italian Renaissance, but it was not a major factor.

The Venetians brought back some significant art pieces from Constantinople. The classical Roman gilded copper statues of four horses that were in the hippodrome in Constantinople for centuries were placed on the top of the St. Mark basilica, and they have been a landmark in Venice ever since. Yet there is no evidence that these specific sculptures had any influence in reinassance art. The equestrian statue of Marcus Aurelius in Rome was instead the one used as a classical model of equestrian sculpture through the middle ages (see the Equestrian Statue of Charlemagne, for example) and the renaissance. Another great example of Venetian loot from the fourth crusade is the porphyr statue of the Four Tetrarchs, which is considered one of the great examples of early medieval art, with the four emperors no longer shown as realistic and beautiful as gods, as was typical of classical Roman and Greek art, but as a set of visual symbols, the swords and crowns representing the imperial authority and the beards on two of the four representing the higher rank of the senior emperors. Again, this and other looted pieces of art did not really influence the art of western Europe in the late middle ages and the renaissance.

A more significant influence from the fourth crusade was the influx of Greek Byzantine artists who moved to the west. Vasari, in his Lives of the Most Excellent Painters, Sculptors, and Architects, begins his narration talking about Cimabue, who was active at the end of the thirteenth century, a few decades after the fourth crusade. Vasari says that Cimabue was the first to break away from the Greek artists he studied under, and according to Vasari he was the first artist in the long narrative arc that led to Michelangelo. While Vasari writes three centuries after Cimabue, and he has a narrative to sell in his Stories with Byzantine art being his example of mediocre art in contrast to the progression of "most excellent" Florentine Reinassance artists, he makes clear that Byzantine artists were well established and popular in thirteenth century Florence. These artists were, in a sense, the first professional lay artists of the Italian renaissance, after centuries in which art was primarily made by the clergy, for the clergy.

This diaspora of artists from the Greek east to the west was not something specifically tied to the fourth crusade. Byzantium had been in decline since the eleventh century, once the empire lost a significant source of tax revenue when the Seljuk Turks conquered most of central Anatolia (which was the reason emperor Alexios I Komnenos asked the west for military help, which in turn led to the first crusade). During the fourth crusade the crusaders were promised significant rewards in exchange for helping with regime change, but the money simply was not there. Byzantium was no longer as wealthy as it used to be, and its artists had already started moving to places where they could be well paid for their art. Venice shows the impact of byzantine art, for example in the mosaics inside St. Mark, but you have examples of Byzantine art elsewhere in Italy in the central middle ages, for example in several great churches in and near Palermo, which were built and decorated at a time when Norman Sicily was among the wealthiest parts of western Europe.

While the influx of Greek art and artists both before and after the fourth crusade may have been a factor in pollinating the early art of the Italian renaissance, economic factors were also very prominent. The thirteenth century was the third century of great economic expansion in western Europe, and the overall wealth of Italian cities and of their merchant class is what allowed artists to be well paid, and more innovative art to be made.

Sources: For primary sources, in addition to the images I linked to, the first chapter of Vasari - Lives of the Most Excellent Painters, Sculptors, and Architects gives a good summary of the "state of the arts" in post-fourth crusade Florence.

Wickham - Medieval Europe has an excellent chapter on the Fourth Crusade and the earlier decline of the Byzantine Empire, and a chapter on the great economic expansion in western Europe between 1050 and 1300. For more in depth coverage of the economic factors behind the art of the renaissance, I recommend Goldthwaite - The economy of renaissance Florence.