As I've said months ago: Burks is clearly the odd man out. Nevertheless, I don't think the Jazz management will trade him right away. They'll probably give Burks some burn no only to increase his value, but also not to risk that the fanbase ends up thinking that one of the players was dumped just for being injured, without being given any chance to prove himself. By not trading him right away, they will also appear "fair" to the other players on the roster - which is a big bonus.
Regarding Burks' minutes, I think they will oscilate between 8 and 20, depending on the match-ups and on how he plays during his first minutes. I think about 75% of his floor time will be spent at the 2, backing up Hood, and the rest eating up a few minutes from Mack and Exum.
According to basketball reference, Ingles currently plays 26% of the time at the two - which amounts to around 4 and a half minutes per game. Those minutes will probably go to Burks. If Ingles remains hot, he'll probably eat up a couple of more minutes at the 3 spot, from 10 minutes per night to about 12. No big deal. Hayward might lose a couple of minutes (especially after Hill returns), which will help him to remain fresh for the playoffs, and Johnson playing time might decrease from 24 to around 18.
The challenge for Snyder will be to keep everyone happy with the reduced minutes. I think he can do it if the team keeps winning...