A creature must usually take a move action to retrieve a stored item like a potion
Like this fine answer mentions, a creature typically takes a move action to manipulate an item, and that "includes retrieving or putting away a stored item" (Player's Handbook 143), like a potion stowed in a backpack or belt pouch. Worse, according to Table 8–2: Actions in Combat, a creature that tries to "retrieve a stored item" provokes an attack of opportunity from each foe threatening the creature.
Potions aren't weapons… unfortunately
To clarify, on draw or sheathe a weapon says that a creature that possesses a base attack bonus of at least +1, during a move action in which the creature moves up to its speed, can also take a free action to draw a weapon (142). In addition, "[i]f you have the Two-Weapon Fighting feat…, you can draw two light or one-handed weapons in the time it would normally take you to draw one" (ibid.). Moreover, the feat Quick Draw (PH 98) allows a creature to "draw a weapon as a free action instead of as a move action." Unlike Pathfinder, it's been my experience that this typically includes splash weapons (e.g. a flask of alchemist's fire, a flask of holy water) yet not potions.
Exchanges in the Dragon #328 Sage Advice column “Official Answers to Your Questions” (80) clarify that the Quick Draw feat's benefit extends to wands, rods, and staffs. (These exchanges are repeated nearly verbatim in the D&D Frequently Asked Questions (68) with which some take issue.) The Rules Compendium repeats this information and clarifies further that "[w]ands are light, and rods are one-handed" (9).
In sum, potions aren't mentioned as a possibility for faster retrieval either by draw or sheathe a weapon or the feat Quick Draw and, while later material expands their mandates, their mandates weren't ever expanded specifically to cover potions like they were to cover wands et al. In fact, this DM's never considered potions weapons, and this player's never had a DM who has. Nonetheless, I can imagine a house rule—especially in, for example, a low-magic campaign wherein every PC is mundane combatant—that says Potions are treated as weapons for the purposes of draw or sheathe a weapon.
Ways to retrieve and consume a potion more quickly and sometimes without risk
It seems to this reader that the game later understood that, as presented in the core rules, potions are pretty terrible magical spell delivery devices. So, largely outside the core rules, several options make potions… not good, really, but, at least, better.
I've mentioned the best potion delivery device—the ready-drink helm (or, reskinned for steampunk, the dwarven injector) (Dragon #294 80–1) (either is 150 gp; 2 lbs.)—in other answers here and here, respectively. Either is a mundane game-changer, giving mundane creatures in particular a significant edge in combat. Either allows a wearer to take free actions that don't provoke attacks of opportunity to consume a potion but there's some danger to the potions the item contains if the wearer is tripped or otherwise falls.
The mundane item potion belt (Forgotten Realms Campaign Setting 96) (1 gp; 1 lb.) or masterwork potion belt (Forgotten Realms Campaign Setting 96) (1 gp; 1 lb.) stores 6 potions or 10 potions, respectively. A wearer can 1/round take a free action to retrieve a potion from either potion belt. (This DM would only allow a creature to wear one at a time, and the creature must pick which to wear.) Although this still provokes attacks of opportunity as normal, it is faster.
The new ranger equipment tactical vest (Dragon #323 103) (5 gp; 2 lbs.) allows a wearer to take a move action that doesn’t provoke attacks of opportunity to retrieve items stored within. The tactical vest holds 10 lbs. of items. (That's 100 potions… or your entire fortune in chalk, sewing needles, and signal whistles.)
The goods item potion bracer (Sharn: City of Towers 159) (100 gp; 1.5 lbs.) holds 3 potions, each of which the wearer can take a standard action that doesn’t provoke attacks of opportunity to consume. A sunder attempt can be made against a potion bracer (AC 14 + the wearer’s Dexterity modifier; hardness 8, hp 5), and an attack dealing more than 4 points of damage destroys a potion. A potion bracer's wearer has a 5% arcane spell failure chance.
The weapon modification hilt hollow (Dungeonscape 30, 33) (200 gp; 0 lbs.) can be added to any weapon. A creature can take a full-round action to load the hilt hollow with a little item like a potion. A creature that doesn't know about a weapon's hilt hollow that succeeds on a Search skill check (DC 30) locates a weapon's the hilt hollow. A creature wielding the weapon can take a swift action to eject the hilt hollow's contents into an empty hand.
The armor, shield, or weapon modification oil chamber (Du 30, 33–34) (1,000 gp; 0 lbs.) can be added to any weapon or shield and two can be added to armor. A creature can take a full-round action to load the chamber with an alchemical or magical oil. The wielder, user, or wearer can take a swift action to apply the loaded alchemical or magical oil to the respective armor, shield, weapon, or weapon's ammunition.
Although only really good for magical oils (that have their own problems1), oil of bless weapon (1st-level spell at caster level 2) (100 gp; 0.1 lbs.), for example, is almost always a welcome addition to a warrior's arsenal.
The prestige class alchemist savant (Magic of Eberron 53-7) can brew a spellvial that has a price 50% higher than a potion and a more limited range of spells it can contain. However, spellvials "can be used as thrown weapons," so a creature that possesses the feat Quick Draw should be able to take a free action that doesn't provoke attacks of opportunity to draw one. Despite this, there's no indication as to how a creature can employ a spellvial on itself, the author seeming to imagine a spellvial being used exclusively on another. (This DM recommends a house rule saying that A creature can make an attack that doesn't provoke attacks of opportunity to use on itself a spellvial.)
In the core rules, Heward's handy haversack (Dungeon Master's Guide 259) (2,000 gp; 5 lbs.) allows a creature to take a move action that doesn't provoke attacks of opportunity to retrieve any item from it. (The DM may permit a bag of holding (DMG 248) (2,500+ gp; 15+ lbs.) that isn't loaded beyond standard backpack capacity to have this same ability, although this isn't mentioned as a bag's "even greater power" like it is with the haversack.)2
In short, Dumpster-diving makes potions sort of bearable. They remain substandard options compared to, for example, wands and lack the versatility of, for example, attuned gems (Magic of Faerûn 21), but they can be forced to be useful, especially if the DM approves the dwarven injector.
1 In what this reader is certain is an oversight, the feat Brew Potion (PH 89) allows creating only potions and not magic oils, perhaps making all magic oils the product of the long-forgotten Age of Inference?
2 Heward's handy haversack in dnd-3e allowed "[r]etrieving any specific item from a haversack [as] a free action[, and t]he handy haversack and whatever it contains gain a +2 resistance bonus to all saving throws." These features were altered and removed, respectively, by 3.5e revision, which, in the first case, is probably a good thing. It was kind of nuts.