No, you don't want to up the saturation for the web. However, it does tend to look that way.
Here's an example from a friend's wedding. The first was simply saved as a jpg in adobe rgb format. The second was converted to sRGB and saved for web.
Now, in a color managed application like photoshop, the two images look almost identical, but in an application like a web browser which (on a Windows PC) isn't color managed the sRGB file looks much better. The Adobe RGB image looks flatter, less saturated, less contrast. That's why you want to convert to sRGB for web use. Also, the save for web stips out all of hte extra stuff stored in the image file, so you can get a higher quality jpg for the same file size.
Note: The only difference in these two images is the conversion to sRGB and save for web on the second image. All the other adjustments and editing are identical.
Dan