I wouldn’t think so. GC is handled by the runtime. It may do a better job of some inlining and code optimization wich would (slightly) improve performance and memory usage but wouldn’t impact the GC.
The .NET 4.6 beta for the Unity Editor though should be a good indication of work and progress being made. That will be a significant update once the Engine utilizes it and will result in a newer runtime and better GC as well.
As @Dustin-Horne says - the new compiler can generate better IL than before in some situations (a mix of more optimized instructions + using newer instructions that weren’t fully implemented in the older compiler), which means those situations might execute faster or with fewer allocations than before.
And as @Ironmax says, it is also still possible to write very bad code that even the new compiler cannot help you with