Burgener's lifters do not participate in CF training. It's 100% lifting, and he'll be the first to tell you that CF is detrimental to lifting performance (as would be any other non-lifting-specific training with any kind of real physical demands--not meant to pick on CF here). He really only uses CF-style training with his high school PE kids or non-weightlifter athletes. His WOD is a clear example of this.
I hesitate to pick on him, but Josh Everett is a good example of the effects of CF training on lifting. Josh is a remarkably strong guy (split snatches 115, split clean and jerks 160 or so I believe). But the more CF he does, the worse his lifts get. He's simply not able to hit the numbers he could before. It's a conscious decision he's made, of course, to place priority on CF over lifting.
Hopefully Burgener and Josh will chime in here so I'm not speaking for them entirely.