this is an excellent theory, but i would imagine searching forward of their last known trajectory would also be an obvious priority for experienced search and rescue teams.
Also, a fire that gave them no chance to even declare an emergency would likely have had to have been large enough to cause a near simultaneous failure of critical systems while somehow never being heard, seen, or (most strangely) smelled until those failures occurred. Also, did the original traffic controllers (or anyone) actually observe the plane as it lifted off? if so, if a tire had lit, they would have seen it due to the extreme wind flare and night conditions.
Also, if the ascent to 45,000 happened after the plane went dark, that's not really consistent with a fire, where i'd imagine you'd be wanting to maintain alt then start your descent ASAP (i don't know this for sure though, and there could have been extenuating circumstances where they anticipated they would need the extra altitude as the article mentions--or maybe with no instrumentation they just got disoriented.) Going that high to try to slow the fire and buy some time, as the article asserts, seems plausible, but any rapid descent would likely be due to a stall, not an attempt to blow out the flames, as without instruments that would be a great way to over speed and rip your plane apart.
rubber creates a thick, noxious smoke which i'm having trouble wouldn't have worked its way in trace amounts into the cockpit and/or cabin and been detected, allowing for some message to the tower. if the smoke didn't circulate at all in the plane then the fire was contained and should have put itself out before burning much of the tire (again that's a guess, but seems like common sense). also a burning tire implies a fire well on it's way, not some smoldering ticking time catastrophe, so how did the plane get as far as it did w/o problem.
also a fire would have the plane veer off course and crash before the fuel was exhausted, whereas the linked article hypothesizes the plane continued on a constant heading until crashing into the indian ocean (after speculating about a fire...the author isn't being consistent.)
Those are the questions that jumped into my mind but it's still a very good hypothesis, just still relies on a lot of assumptions and the more i think about it, having seen nearly every air crash documentary, doesn't make sense as is and will need some modifications.