If you are writing a Web page, you should understand how people use the Web, and one of the keys to the Web is links. If your links are not distinctive and stand out from the surrounding text, they will not be useful to your readers.
The most common way to ensure this is to leave the underlines on the links. Removing the underlines makes it more difficult for your customers to find the links.
The converse is also true. It's not a good idea to underline other elements of text within your Web pages. People understand links to be underlined text, and if you have additional underlined text this will just frustrate people.