I'm not sure that Gillislee's fate would have much to do with Lewis or Burkhead, although Lewis and Burkhead would seem to affect each other, at least a bit, in that they may provide each other with some negotiating leverage. The Pats would seem more likely to accept the loss of one of them than both of them. Of course, that may work in reverse, too - in a "first one who confesses gets less jail time" sorta way.
Anyway, cutting Gillislee saves a net of only about $1.7M, and leaves them short a decent RB with NFL experience, and I don't think it helps all that much with the other two.
Just as a point of age/contract length reference...
... White just turned 26 a couple days ago
... Gillislee turns 28 on Nov. 1st (already under contract).
... Burkhead turns 28 on July 2nd
... Lewis turns 28 on Sept. 27th
Deals may max out at 3 years for Burkhead and Lewis (through age 30 for both), and would like be based in per-game roster bonuses (for health).
I'm not sure that the Pats would offer Burkhead a huge amount more than what White signed for (which was 3 yrs/$12M, with $4.7M guaranteed) - maybe up to 3 years at $4.5M per (Lynch/CJ Anderson level) - and I'm not sure that he'll get many (if any) offers much better than that.
Lewis, to me, seems more likely to see outside offers significantly above what Burkhead might - and possibly more than what the Pats would offer - maybe a bit ore than DeMarco Murray money ($6.375 APY, $9M guaranteed). I could see Lewis going to the Colts with McD for that.
My guess is that Gillislee stays, at least through Camp, Burkhead gets re-signed, Lewis walks for a bigger payday, and the Pats backfill the 90-man roster with possibly a lowball UFA or SFA (outside chance of another trade or RFA tender), a couple UDFAs and possibly a late-round pick.