How can I extract color from food ingredients that won't add flavor when making mocktails?

It is basically impossible to do that in home conditions.

Assuming that you are starting with the whole vegetable (or fruit, or flower...), the closest you can do is to turn it to juice. Actual extraction is almost always futile - the solvents in the kitchen are limited to fat, water and alcohol. Fat is completely out of the equation, both because it won't extract much color and because it won't taste good in a cocktail, a water extract would be inferior to a juice (which is already water-based), and for alcohol extracts, look at traditional homemade liqueurs - besides getting a lot of taste along, you will also get colors in the brownish range. Pale pinks or yellows could be possible too.

The juice method will give you a decently colored liquid, but with the caveat that you won't be able to concentrate it or preserve it. So, if you want a green cocktail, you may have to mix it with equal parts spinach juice to get a good color. Also, you will face an unpleasant tradeoff between color and texture - the more you filter the juice, the paler your color. Also, expect a shelf life of minutes to hours before the juice starts browning.

If you are really OK with something industrially produced, such as the powders Megha mentioned, you can of course try them, but then again, you could just go for industrially extracted food coloring.

I know you can color foods with natural-dye ingredients, like using spinach powder for green, beet powder for red, and so on. These will change the taste a little, but it doesn't have to be much (especially if you can keep the color powder on the surface).

I realize you're concerned about the taste difference, well, I've heard of cupcakes made with spinach powder (or possibly the frosting was made green, not sure), that were still good... the powder was barely noticeable, the amount needed even for a good color didn't change the taste much at all. So it is possible, if you're not using heaps of powder, for the difference in taste to be subtle or unnoticeable, especially if you've other strong flavors in there.

I know there are places where this kind of natural food-coloring-powder is sold, if you don't want to dry and grind your own (and better control in said drying and processing will give better color for less). I found one such set on bluechai's website, there are likely to be others

"natural" green food coloring is chlorophyll. Try putting some spinach into a blender, and letting the result drip through a coffee filter.

For purple, boil red cabbage in water, then strain and use the boiling liquid. Note that the precise color will be affected by the acidity of the drink.