Would a fighter wielding a torch in his primary hand and holding a large shield in the other hand be allowed to perform an Attack of Opportunity with the torch from an opponent who provoked one in the area he threatens by trying to move past him?
The typical not-flat-footed fighter can make an attack of opportunity with the torch in this situation
A typical Medium creature that's carrying in one hand a torch and employing in its other hand a readied heavy shield and that's not flat-footed typically threatens all of its adjacent squares with both the heavy shield and the torch and is capable of making one attack of opportunity with the creature's choice of either the torch or the heavy shield.
Unless the DM rules otherwise, a creature wielding an improvised melee weapon threatens an area just like the creature would with a normal melee weapon. Further, the torch has actual rules supporting its use as an improvised weapon. A creature that's carrying a torch can "treat it as a one-handed improvised weapon [i.e. typically suffering a -4 nonproficiency penalty on attack rolls with it]… that deals bludgeoning damage equal to that of a gauntlet of its size [i.e. 1d3 points of bludgeoning damage for Medium creature], plus [presumably if lit] 1 point of fire damage" (Player's Handbook 127 and links mine). (Note that while a torch deals damage as if it were a gauntlet, a torch—fortunately—shares none of a gauntlet's considerable quirks!)
A creature that's carrying a heavy shield can make a shield bash attack. While shield bashes mentions using a shield as an off-hand weapon, a shield can also be used instead as a creature's main weapon à la Captain America and the D&D Main FAQ (52). (I'm certain a more authoritative source for shield bashing as a primary weapon exists elsewhere, but the FAQ's answer is convenient, accessible, and—I believe in this case—accurate.) However, making a shield bash—even as an attack of opportunity—causes the typical attacker until the attacker's next turn to lose the shield's shield bonus to AC.