Michael Barber argues in "Jesus as the Davidic Temple Builder and Peter's Priestly Role in Matthew 16:16-19," JBL 132 (2013): 935-953, which you can download here, that Matthew portrays Jesus as the new temple builder. Here is the abstract:
It is widely accepted that Matthew presents Jesus as fulfilling Jewish eschatological expectations, particularly, Davidic hopes. However, although Jesus frequently speaks positively about his disciples' participation in the cult in Matthew's narrative (e.g., 5:23-24), little attention has been paid to Matthew's interest in Christ's fulfillment of the cultic dimensions of future hopes. In fact, ancient Jewish sources repeatedly express not only the belief in an eschatological temple but also expectations of a reformed and/or new priesthood. In this article, I argue that such hopes inform Matt 16:17-19. I begin by arguing that Jesus' building of the church is best understood in light of Matthew's Davidic Christology, an aspect of the evangelist's portrait of Jesus that many scholars have noted. Specifically, building on the work of others, I contend that Jesus' response to Peter's confession involves allusions to Davidic traditions of temple building (cf. 2 Sam 7:12-13; 1 Chr 17:7-10) (e.g., Ådna, Meyer, Wright). Going on, I demonstrate that Jesus' description of Peter's role in the following verses seems to portray him as one holding a priestly role. In particular, as many scholars have noted, Jesus' words to Peter appear to echo the description of Eliakim in Isaiah 22 (e.g., Davies and Allison, Hagner, Willis). What is frequently overlooked is that this passage describes Eliakim as wearing garments usually associated with the high priest (cf. Isa 22:21 with Exod 28:4), an aspect of the passage not lost on Jewish readers (e.g., Tg. on Isaiah; b. Ta'an. 29a). "Binding" and "loosing" also seem linked to responsibilities typically associated with the priesthood (teaching, judging, mediating divine forgiveness). In light of this I argue that this passage provides Jesus with the perfect quarry: if the church is a temple, its leadership is naturally described in terms related to priestly responsibilities.
