No the PC doesn't need to be on. However, wireless adapters can have a tendency to "lose" their IP address, the router is unaware that its client has lost the address and it isn't renewed. It is necessary to "Repair" the connection to force a renewal of the IP address. This can be caused by interference from other wireless devices such as cordless or mobile phones, microwave ovens and so on. Placing either the Wireless Router or the wireless adapter close to large metal objects or large volumes of water, in particular central heating radiators, can cause problems with the radio signal.
Sometimes the situation can be improved by giving the Wireless Network connection a fixed IP address in the same range that the router operates in.