Not at all. Doesnt have to be IP, and doesn't even have to link computers.
Routers as opposed to switches and bridges and hubs, make decisions based on a higher layer in the packet switching protocol than the raw layer at which their interfaces act.
I.e an ethernet hub makes no decisions..it broadcasts all traffic on all ports.
A switch builds a table of MAC level addresses and learns which port to send the packet down.
A bridge is similar to two switches connected by something else than an ethernet cable. It learns which mac adresses are where.
A router, on the other hand, makes decisions based on a higher level protocol. Typically IP, but not necessarily. It has routing tables that are either preset, or downloaded dynamically that are used to make those decisions. It does not 'learn'. It is 'told'.
A typical DSl 'router' conatins the equivalent of an IP router, an ATM half bridge, an ethernet port hub, or switch, and a DSL modem, plus a NAT and firewall, and possibly a wireless half bridge or hub as well.