Story

There will always be a need to have a basic method for accessing information.

Web browsers can degrade nicely. Standard HTML can be displayed and purposed by modern computers. A mobile device with a 7" screen is capable of showing web pages just as well as a laptop. The main difference is in the UI.

We still have a range of protocols and tools which could be described as being redundant or well past their use by date, yet we still use them. FTP, telnet, http (as opposed to https or spdy), vi/notepad, text passwords all have place. Nothing is going to be thrown away just because the new shiny way of doing things has come out. Some countries still use older technology like cassette tapes.

So, the answer is no. Web browsers are not dead on mobile phones. If recent improvements like HTML5 with native video support are any sort of indication then they definitely will not be going away soon.