I think this idea that Kane will still be the same player after being refused a move is a bit misguided. Sure, he won't down tools because he's a consummate professional, but a huge part of motivation and desire is subconscious and essentially outside of your control. So without trying to do anything different, I don't think he'll be able to keep putting in 100% effort. There's a degree of almost faking it involved.
Just imagine if you weren't allowed to leave your job. If the pay is great, you feel valued, and you feel like you owe something to your employer, you're probably going to continue trying your best anyway. But are you really going to tell me that you'd still have the raw intrinsic hunger that came naturally and inevitably when you actually wanted to be there?
If Kane is kept, I can totally see next season ruining his legacy at Spurs and tanking his relationship with the fans, which would be tragic. If the team doesn't do well - and I don't think there's any reason to believe they will - every single mistake Kane makes will be replayed ad infinitum, overanalysed, and used as evidence that he doesn't care enough because he doesn't want to be at the club. Every lull will be perceived as laziness. Even the most typical, garden variety mistakes will be pounced upon. Kane's stamina management protocol already makes him look low-energy at times in games, but next season it'll be scrutinised anew and declared further evidence of a decline.
I'm sure Kane will continue to do the best he can, at least on a conscious level (as above, I think it's impossible to give 100% under the circumstances, no matter how much you tell yourself you want it). It'd be harsh to judge him for his effort if he's doing as much as his body and brain will allow. Anyway, the confluence of all of these factors - Kane's subconscious loss of edge, a few lean patches, another bad season for the team - could so easily lead to a rift between him and the fans. I can see it being a horrible break-up and that's sad.