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While potions are very magical in their action, the actual technology of making them seems more like chemistry than magic, assuming one has access to already-provided magical ingredients.
If that's the case, can a Muggle who found/was provided with all the right magical ingredients and the recipe actually make a magical potion?
As per Rowling herself, the answer is mostly "no".
Q: Can muggles brew potions if they follow the exact instructions and they have all of the ingredients?
J.K. Rowling: Well, I'd have to say no, because there is always... there are magical component in the potion, not just the ingredients. So, at some point they will have to use a wand. I've been asked what would happen if a Muggle picked up a magic wand in my world, and the answer would probably be something accidental... possibly quite violent. Because wands, in my world, is merely a vehicle, a vessel for what lies inside the person. There is a very close relationship -- as you know -- between the wand that each wizard uses and themselves. In fact, we'll find out more about that in book 7 (crowd applauds).
For a muggle you need the ability, in other words, to make these things work properly but you're right and I think that's an interesting point. Potions seems, on the face of it, to be the most Muggle-friendly subject. But there does come a point in which you need do more than stir.
Source: An Evening with Harry, Carrie and Garp: Readings and questions #1, August 1, 2006, Radio City Music Hall, New York, NY.
I've been asked what would happen if a Muggle picked up a magic wand in my world, and the answer would probably be something accidental... possibly quite violent. Almost makes it sound like muggles can use magic, but not in a controlled way. As though wizards are just better at controlling/focusing it.
Allow me to go off-topic a bit, but I would like to explain why Squibs can't make Potions and then draw some conclusions.
In Chamber of Secrets, Harry sees Filch's Kwikspell course letter. Remember that Kwikspell is a course designed to try to teach Squibs spells (a course that doesn't work JKR has said). When we see Filch's Kwikspell letter, a "testimonial" from Madam Z. Nettles of Topsham states that:
“I had no memory for incantations and my potions were a family joke! Now, after a Kwikspell course, I am the center of attention at parties and friends beg for the recipe of my Scintillation Solution!”
This seems to indicate that Squibs can't make Potions after all. and If Squibs can't, Muggles can't either.
Also, I would like to emphasize on the fact that Potions isn't all just chopping, slicing and stirring. There has to be a magic component to it beyond the use of magical creatures for ingredients. So I reckon that whilst making a potion it must continuously draw on your magic to at least some degree.
For simpler potions (going by the books thats years 1-6 at least) it must happen pretty much automatically. For more complex potions, such as the luck potion and so forth, there is perhaps a technique that Masters use to add their magic in the correct places in the correct strengths.
As revealed on Pottermore, although potion-making doesn't appear to have a great deal of magical input, the reality is that at some point in the process wandwork is essential to add magic to the ingredients and that even adding magical ingredients to a mixture isn't sufficient to create a working potion.
"It is often asked whether a Muggle could create a magic potion, given a Potions book and the right ingredients. The answer, unfortunately, is no. There is always some element of wandwork necessary to make a potion (merely adding dead flies and asphodel to a pot hanging over a fire will give you nothing but nasty-tasting, not to mention poisonous, soup)."
Since a muggle is incapable of performing wanded magic, they would by definition be incapable of brewing a magical potion.
As DVK has referenced in his answer, this information was already largely confirmed in an interview in 2006:
Q: Can muggles brew potions if they follow the exact instructions and they have all of the ingredients?
JKR: Well, I'd have to say no, because there is always...there are magical component in the potion, not just the ingredients. So, at some point they will have to use a wand. I've been asked what would happen if a Muggle picked up a magic wand in my world, and the answer would probably be something accidental... possibly quite violent. Because wands, in my world, is merely a vehicle, a vessel for what lies inside the person. There is a very close relationship -- as you know -- between the wand that each wizard uses and themselves. In fact, we'll find out more about that in book 7.
For a muggle you need the ability, in other words, to make these things work properly but you're right and I think that's an interesting point. Potions seems, on the face of it, to be the most Muggle-friendly subject. But there does come a point in which you need do more than stir.