No, but OpenGL might ;)
The base Vulkan specification only supports SPIR-V. However, Vulkan does allow for vendor extensions. And NVIDIA is already on-record on this matter; they will be providing a Vulkan extension to be able to shove GLSL into their Vulkan implementation.
That shouldn't be taken to mean everyone else will.
However, let's not forget that Vulkan is not going to magically appear everywhere, nor will OpenGL instantly vanish into the aether the moment the Vulkan spec ships.
First, there are entire classes of users who will gain virtually nothing from switching to Vulkan. Who are getting good enough performance for their problem domains from OpenGL, so they don't need to. Pretty much every 2D game you've ever seen has little to gain performance-wise from Vulkan. Even the CPU power consumption gains from VK are minimal for them.
Second, users who would benefit from Vulkan will still take several years to switch over.
Now, if you want to understand my cryptic comment, it is entirely possible for OpenGL to start being able to take SPIR-V as its shading language. SPIR-V contains virtually every feature of OpenGL 4.5, with the exception of shader subroutines (a feature so terrible I won't even link you to the OpenGL Wiki article for fear that you might try to use it).
Indeed, if SPIR-V were only intended to be used for Vulkan or OpenCL, there would be no reason for it to have so many GLSL-based decorators and such in the language. Vulkan uses descriptor sets, not binding points, so why does SPIR-V allow you to specify binding points for textures/images/etc? The same goes for many other GLSL-isms in SPIR-V.
All it would take is an OpenGL extension to allow SPIR-V to be fed to
glProgramBinary, likely with a dedicated format enumerator for it.
Will Vulkan kill GLSL? No. But OpenGL and Vulkan might. Though even then, it'd take a long time for it to die.