There are various options that let you make your enemy drop a held object (for example, the "Disarm attack option").
Would a character that just disarmed an opponent be able to pick up the dropped weapon from the feet of the target?
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So long as you are within reach of the object on the ground, you can pick it up. Though this action will cause you to take an attack of opportunity.
The sleight of hand action allows you to take unattended objects from creatures within reach. If both creatures are of medium size, they cannot occupy the same 5ft square. So the implied rule here is that the target creature must be within reach (adjacent square) in order to pilfer something off their person. Theoretically you could steal a boot dagger, I.E. you bend over, but even in that instance you cannot occupy the same space.
As a DM, so long as the object falls in front of the creature and is between the two of you, you can take the attack of opportunity (you bent over) and grab the object off the ground. If the creature is in front of the object, you would need to reposition or move through the space (tumble check) in order to grab it.
The special attack disarm says that if a creature uses a weapon to successfully disarms an opponent, that creature "knock[s] the opponent’s weapon out of [its] hands and to the ground" (Player's Handbook 155). This is clarified further by the Rules Compendium on Resolving a Disarm that says that a successful disarm leaves "[t]he weapon… on the ground in the defender’s space if [the creature that performed the disarm successfully] used a weapon for [its] disarm attempt" (45).1
While most actions that would see a creature entering or reaching into a foe's space do provoke attacks of opportunity (e.g. the typical creature making a normal unarmed attack or the special attack grapple), the Player's Handbook on Attacks of Opportunity says that a creature can't "make more than one attack [of opportunity] for a given opportunity" (137). The Rules Compendium on Multiple Attacks of Opportunity expands on this idea:
Some abilities allow you to make more than one attack of opportunity per round. Most such abilities, unless they say otherwise, don't let you make more than one attack for a given opportunity. If the same opponent provokes two attacks of opportunity from you, however, you could make two separate attacks of opportunity. Each provoking act represents a different opportunity. (19)
Taken together, this means that when a creature takes a move action that provokes an attack of opportunity to pick up an item (as per Table 8–2: Actions in Combat (PH 141)), that act of picking up the item provokes a lone attack of opportunity from each foe that threatens the creature even if the creature is reaching into a foe's space to pick up that item.
To be clear, a creature can't pick up an item if the item isn't within the creature's reach (e.g. a typical human that disarms a foe 30 ft. away by employing the feat Ranged Disarm (Complete Warrior 103) can't usually then take a move action to pick up the foe's weapon). Similarly, a creature can't pick up an item if the creature has no manipulators available to pick up the item (e.g. a typical human that disarms a foe but maintains her two-handed grip on her appropriately-sized longspear can't usually then take a move action to pick up the foe's weapon).
1 This DM rules that the creature that successfully performed the disarm picks the square if the foe occupies multiple squares. Ask your DM.