I used to use a Peterson Strobe Tuner, like most have here. I went to a Boss TU-2 years ago for "on the fly" tuning from the stage front as it fit right into my pedalboard. We never had a tuning issue onstage between those using rack tuners (Korg) and the TU-2. Since going to the Axe Fx I have taken the TU-2 out of the loop and remain perfectly in tune with others. Now adding the MFC-101 (which shows tuning on the display) I will be back to having accurate tuning at my feet, where I can see it during performance. And, as noted, the Axe tuner can be adjusted in case your band tunes down, etc, for live shows.
For me, having another tuner would be redundant. But then, I have seen people with stompbox effects in use instead of the same ones in the Axe Fx, which are superb. Sometmes performances come down to being comfortable and confident you have what you need in the chain whether it makes sense to anyone else or not.
Reader's Digest version: You don't need another tuner, but it won't hurt if one makes you feel better when performing.