I honestly cant read the Lewis/Burkhead situation at all. Like I said before Burkhead was a total BB move so I am not sure if Josh would actually be a big champion of his. And Lewis really made it sound like he wants to be here. I get that money ultimately talks but I could see them extend both, letting Gillislee go and drafting a big back. We are relying on those RB/LB matchups way too much to try to save money on receiving backs just to get screwed by one injury.
It's hard to know if Lewis' success this season was due to McD's play designs or Fears' coaching + Scar's OL. And it's impossible to know how Lewis himself perceives all that. It's also impossible to know what kind of price tag the Pats would put on him repeating his 2017 performance over the next 3-4 years (and whether or not that would be enough to keep him around).
Aside from his injuries, Burkhead worked out really well. His 518 total yards from scrimmage and 8 TDs in just 10 games appears to indicate a significant upside. Regardless what happens with Lewis, I'd guess that Burkhead gets an offer from the Pats that's very similar to what White signed (3 years, $12M total with $4M of that guaranteed), and I think there probably won't be many outside offers for Burkhead for significantly more money than that at this point.
The other side of this for both Lewis and Burkhead going forward is that, in the absence of Edelman and Mitchell (and Hogan for half a season), and in the virtual absence of a second viable target in the TE group, RB receiving targets were way up in 2017 - perhaps inflated above what was originally planned and what might be expected in the future (with healthy WRs and a better pass-catching TE behind Gronk). So, their $ value probably needs to be weighed against that in addition to their 2017 production.
The thing with Gillislee (considered in isolation) is that he put in more work in 9 games with the Pats than he'd put in with the Bills over the entire 2016 season (which, in turn, was nearly double his usage for his entire previous career that began in 2013 with the 'Fins). He wasn't great in those 9 games for the Pats, but he wasn't awful, either. He's signed through 2018 at $2.18M with ZERO dead money if cut.
I'd say that there's a pretty good chance that the Pats acquire at least
some RB competition for the 90-man off-season/Camp roster, whether Lewis leaves or not. However, unless there are negative things going on with him behind the scenes, it makes no sense to me to cut Gillislee before the end of Camp - until or unless some new acquisition outshines him. Bird in the hand and all that.