Is Visual Basic supported by .NET Core on Linux?

No, there is no Visual Basic support yet. At the very bottom of the roadmap in the "future work" section, it said at one time, that Visual Basic support was set for quarter 3 of 2016, but it is still not released as of March 2017. It is still planned.

Since ASP.NET Core is the most common thing that targets the .NET Core framework, the answer I've provided refers to that.

Visual Basic is supported by .NET Core starting with .NET Core 2.0 Preview 1.