Well, it depends on your definition of "resume."
Personally, I don't think resumes will ever become obsolete because I don't consider a resume to be a Word file (or Pages, Google Docs, PDF or piece of paper). I consider a resume to be the information contained in that document - not the document itself.
To me, a resume is a compilation of information that is put together to show a potential employer that you have the necessary qualifications (experience, skills, training, etc.) to provide a positive contribution to their company. It's not a document - it’s a message - and the form in which that message is delivered has changed over the years, and likely will continue to.
Maybe one day we'll only use LinkedIn profiles as resumes, and separate documents will no longer be necessary. Some people probably believe that's the case today. But regardless of whether you use a resume, a LinkedIn profile, or both (highly recommended!), you still need to put out the best message. You still need to present the right information in a compelling way. You still need to attract the recruiter's attention. You still need to make them want to contact you.
Just because you're filling out an online form rather than a Word document doesn't mean the purpose isn't the same.
In reality, both your LinkedIn profile and your separate resume can be considered your "resume" if you're job searching. Both use headlines at the top, both can include a summary, both list your work experience in reverse chronological order, both have room for education and training, and so on. If you're job searching, both serve the same purpose. (Of course if you're not job searching you may not use your LinkedIn profile the same way - but even then, one day you will be so why not maintain it?)
One day we probably won’t use typical “resumes” any more. In all likelihood, everything will be done through online forms. However, the fundamental aspect of job searching – the need to convince someone of your expertise – will never change. Therefore it’s absolutely critical that your message, regardless of the format in which it is delivered, is as powerful as possible.