The 20-year-old has made 14 appearances for Rene Hake's side this season but the 2009-10 champions lie in 16th place and are serving a three-year ban from European competitions.
Tapia, a Peru international, has therefore chosen to make the switch to a Feyenoord side battling for a return to the Champions League in 2016-17.
"We've very pleased to sign Renato Tapia," Feyenoord director Martin van Geel told the club's official website.
"He's a very talented boy, who already has the required domestic and international experience despite his young age.
"He's also a multifunctional player, though we see him mainly as a midfielder. He'll compete for that role at Feyenoord.
"We seem him as both a short and long-term gain - a player for the future who is immediately usable.
"We just have to wait until his work permit is in order, which can take several days. He therefore is not yet available for the upcoming matches against Heerenveen and ADO Den Haag."