I've researched this before and my conclusion was the Spanish Reconquista was probably the most influential event leading to the 15th century Italian renaissance. Whole libraries that the Arabs had were made available to the west, during the 11th century quite a bit of Aristotles works were obtained via the Reconquista. Its interesting that the first European universities, Oxford, Bologna and others were setup then. I don't believe the Reconquista was a bloodbath, many of the Arabs were friendly with the Spanish afterwards. Paper was also introduced via the Reconquista, although Italy also obtained paper making technology via Venice and Genoa's trading contacts with middle east. Gothic architecture probably is derived from Islamic architecture the Europeans saw in the areas taken from the Arabs in Spain. It is interesting that the Reconquista seems to be a more significant event to the French and Spanish, for some reason English speaking history seems to downplay its importance. However, there were various renaissances trying to 'boot up' in Europe. The Carolingian's (Charlemange) had a renaissance in the 700s. There was an Ottonian Renaissance in the 900s, one of the Ottos married a well educated and cultured Byzantine princess. Then there was a 12th century Renaissance which was probably the result of the Reconquista, but got derailed by the black death and other problems. My general opinion is the later Italian renaissance in the 15th century was so successful because of the printing press. It made wide scale circulation and exponential growth of knowledge possible and made sure the it didn't burn out like all the previous renaissances.

This has been asked before, although it seems to lack in answers, but here are some links to the previous threads.

Here's a post from last month, with a single answer: https://www.reddit.com/r/AskHistorians/comments/4lf6uv/to_what_extent_was_the_renaissance_caused_by_the/

A different thread from 4 years ago, with a few more answers: https://www.reddit.com/r/AskHistorians/comments/uks72/how_much_influence_did_the_fall_of_constantinople/

This one isn't directly about their influence in the Renaissance, but it does mention access to Greek texts and how it caused development elsewhere but apparently not in the Empire: https://www.reddit.com/r/AskHistorians/comments/25a93l/the_byzantine_empire_had_access_to_all_the/

Edit: /u/EconomistMagazine take a look at the third link, I think it answers your question.