Lol, no they don't. He would need to be tip-toeing around corners to loose enough heat for that. Not to mention the 1000C brakes just cm's away.
Lateral loads create heat, and he is lapping numerous seconds a lap faster, and specifically cornering faster than HRT and Marussia, but we don't see those teams coming in with ice cold tyres.
Its a far more fundamental issue than just cold tyres.
The heat from the tyres in cornering comes from the a lot from how the car corners. A MUCH better car with MUCH more downforce wouldn't slide sideways, nor push sideways on the tyres anywhere near as much as a worse car with awful downforce. Then there is the simple issue that, he's mentioned tyre heat as a potential reason for being slower, because he's comparing to Hamilton. Tyre heat wouldn't be an issue if Hamilton was exactly as slow as Button... its all relative. A HRT might have zero tyre heat, but be unable to go faster even if the tyres were at the perfect temperature anyway.
The only real comparison is vs Hamilton, who you can physically see is both faster(and not much of that is going to be down long straights with the same engine) and takes a much more aggressive line in corners.
Some drivers take very different lines, but different cars react differently, one car might want to take any easier line a bit faster, another car might react better taking a very tight line slower. All that can really be said is, Hamilton has got the car working, Button hasn't..... somehow its taken well over 2 years and he hasn't tried to copy Hamiltons line over a track and see if its faster.
ultimately I've said before, he doesn't have the balls for it. The best drivers will push the car on just about every corner, fighting on the limit and making adjustments as the front or back slide out of line ever so slightly. Button takes an easier line, a tad slower and not on the limit. I mean we're also basically assuming that with him saying SO many times that the tyre heat is a problem, that someone at Mclaren can physically see his tyre temps are lower.... they of course might not and just don't want to contridict Button publically and allow him to have the excuse.
I think most people can basically see that Button is a less aggressive driver and always has been, and it wouldn't be surprising if that costs you a little tyre temp. I still don't think he's significantly slower than any other season, he's always been slower, but the tyres do seem to stay better, longer, if you can work them well.