How to recolour a section of a path

There are 4 ways.

1. Create Dashed line for a stroke with different Stroke/Gap colors like SJReal suggests. It's good if you need just 2 alternating colors and sertain sizes of parts.

2. Create a gradient with the needed colors like S_Gans says (put stoppers as close as you can to have sharp edges of colored parts) and set it along stroke. Make sure to set all the stoppers in the same color space.

3. Separate filled object without stroke and stroked object without fill and cut ithe last one by scissors to get the needed pieces like Ton describes. The inconvenience is that you will have many separate objects. Don't forget to group the pieces to not lose any. 

4. Create an Art Brush. Draw the set of colored rectangles, convert it into a brush and set to your shape. In this case you can't use gradients, each partial rectangle must have a plain fill color.

There are 2 methods I an think of. One is to simply Copy/Paste in Front. Then cut out the piece of path you want to change (on the top object) using the scissors tool, or simply adding points, and deleting the other points. Remove the fill from that portion and you'll be left with only the stroke over that one part, which you can change to whatever color you want.

Another method may be to Stroke with a Gradient, with the gradient recoloring just that portion of the stroke (this may not work well, depending on the particulars of the piece with which you're working). You can experiment with the gradient placement using the radial or linear gradient for this (linear would affect opposite ends of the stroke, radial could be managed over one portion), and also experiment with the color stops and transitions to get the edges as you want them.

Can you give a better idea of what you are trying to do? Is it something like this?