It depends on the color that you dye your hair. If you have extremely dark brows and dye your hair platinum blonde, it makes sense to lighten them a few shades. Likewise, if you have light blonde hair and dye it black or dark brown, your eyebrows will most likely be far too light for your hair color. Using a bit of dye to darken them a few shades could help them look more natural.
However, there are a few important things to remember. First, you should never dye your eyebrows the exact same color as your hair. A good rule of thumb is that if you have light hair, your eyebrows should be one shade darker than your hair, whereas if you have dark hair, your eyebrows should be one shade lighter than your hair. 
Also, you should never use regular hair dye on your eyebrows unless it is being applied by a professional. These dyes are usually too harsh for the delicate skin around the eye area. Instead, you should look for a brow tint. These are vegetable-based dyes that are a bit more gentle on the skin and are less likely to cause problems. Even using these milder dyes, however, your best bet is still to go to a professional to have them done. They can help ensure that you get the right color and that you don't wind up with a bunch of dye on your skin or a bad reaction that leaves you with a rash.
If you are lightening your brows, you should look for a bleach that is designed for use on the face. Keep in mind, however, that these products are still relatively harsh even though they are made for use on facial skin and should be used with caution -- particularly since you are using them around your eyes.
You can also skip the dye completely and just use eyeshadow, brow color, or a brow pencil to change the color of your eyebrows. You can even lighten your brows a bit going this route by choosing a brow pencil that is designed to stick to both the hair and the skin in a lighter shade than your natural brows.  Although it won't lighten them dramatically, in many cases it may be enough to help them blend more naturally with your hair.
If you dye your hair red, you definitely don't want to dye your eyebrows red as well. Instead, your brows should either be a medium brown or dark blonde shade, depending on the overall lightness of your red hair. Try looking at some photos of celebrities with red hair for inspiration.
If you decide that you want to try dying them yourself, this video has some helpful tips: