Cannot reproduce here (linux, 970m, 435.21 drivers...)

It works fine in 2.80. It does not matter if I start in 2.80 or I start in any 2.81 alpha.
I know there are some changes in the node, but I am not modifying the settings at all. I just insert the node without changing anything and it does not show the environment image and when rotating it does not update by removing the previous images.

I included the screenshots and you can see the nodes that all have default values.

How do I pack those image files into the .blend file? You still can see them, why do you need them packed?
I also used the sunrise environment file that is a file included in Blender. I also tried with external .hdr files and it is all the same.

Did you load the .blend file and tried in both versions?

I also tried with both image texture and environment texture. The problem disappears in 2.81 if I disconnect the output of the vector mapping node into the image or environment texture node.

I packed the image used for world texture into the .blend file. I did it in 2.81 latest. Let me upload it here:

Update. I tried changing the render from eevee to cycles and it works in 2.81. This problem may have to do with some incompatibility with my graphics card and some new changes in 2.81 eevee rendering together with attaching the node mapping. It works fine if I use 2.80 version.

This is doing the same thing for me for the last several 2.81 builds with totally different assets.

Operating system: Windows-10-10.0.18362 64 Bits
Graphics card: Radeon RX Vega64 ATI Technologies Inc. 4.5.13571 Core Profile Context 19.9.2 26.20.13003.1007

Blender Version
Broken: version: 2.81 (sub 12), branch: master, commit date: 2019-09-21 17:21, hash: rBc71be416e09b
Worked: 2.80

Thank you Tom. I also tried on a different computer, a laptop and it also works with Cycles but it does not render correctly with EEvee (in 2.81). It works fine in 2.80.

This seems to be fixed in the current build.