An IP router SHOULD satisfy the Host Requirements with respect to IP multicasting, as specified in [INTRO:2]. An IP router SHOULD support local IP multicasting on all connected networks. When a mapping from IP multicast addresses to link-layer addresses has been specified (see the various IP-over-xxx specifications), it SHOULD use that mapping, and MAY be configurable to use the link layer broadcast instead. On point-to-point links and all other interfaces, multicasts are encapsulated as link layer broadcasts. Support for local IP multicasting includes originating multicast datagrams, joining multicast groups and receiving multicast datagrams, and leaving multicast groups. This implies support for all of [INTERNET:4] including IGMP (see Section [4.4]).
Although [INTERNET:4] is entitled Host Extensions for IP Multicasting, it applies to all IP systems, both hosts and routers. In particular, since routers may join multicast groups, it is correct for them to perform the host part of IGMP, reporting their group memberships to any multicast routers that may be present on their attached networks (whether or not they themselves are multicast routers).
Some router protocols may specifically require support for IP multicasting (e.g., OSPF [ROUTE:1]), or may recommend it (e.g., ICMP Router Discovery [INTERNET:13]).