In short, probably not. Clipping paths have been around for decades. In the 1980s and 90s, they were the only way to cut an image out from its background. That changed in late 1999, when TIFF files gained support for transparency. In 2002, design software like InDesign and QuarkXpress began fully supporting Photoshop files too. Clipping paths are only ever used for print. You'll never need one for a file that will be used on screen.
The main problem with clipping paths is that their edges are razor sharp. Magnify a printed page and you can see the tiny printing dots sliced cleanly through. If part of your image is soft or out of focus, a hard-edged path can look out of place, especially against a dark background. This can make the image look dated. A better option is an image with a transparent background, which keeps the photo's natural softness and depth.
So why do people keep telling you that you need one? Bluntly, the person saying so often doesn't know what they're talking about. Many people are afraid to ask questions in case they look foolish. This mistake can cost you real money, because clipping paths take time to make, and the more complex the path, the higher the price.
I wrote this blunt post for two reasons: to save you money, and to save me time (which also saves you money). I've worked in digital design for over 30 years, and I've seen too many people waste hundreds of pounds because they were afraid to ask questions.
Next time someone says you need a clipping path, tell them you don't, and keep the cash.
If you’re ever unsure of which questions to ask feel free to get in touch.