Hi,
Yes you would expect interference between multiple IR remotes. If you used Arduinos to build the remote controls as well as the Robots, you could assign different codes for each robot and the others would ignore others. But you still have real-time interference when multiple remotes transmitted at once.
See: http://arduino-info.wikispaces.com/IR-RemoteControl
The lowest-cost RF remotes are nRTF24L01. See: http://arduino-info.wikispaces.com/Nrf24L01-2.4GHz-HowTo