No! The type of mobile carrier you use won't have any effect on your use of the Fi collar or app.
With an active subscription, the Fi collar will act as its own cellular device on the AT&T LTE-M network. Your smartphone can be on any cellular network - it does not need to be on AT&T for the Fi collar to function.
Note: Without a subscription, AT&T LTE-M network connectivity and 'Lost Dog Mode' are not available.