System Information
Operating system: Windows-10-10.0.19044-SP0 64 Bits
Graphics card: NVIDIA GeForce RTX 3080 Ti/PCIe/SSE2 NVIDIA Corporation 4.5.0 NVIDIA 512.15

Blender Version
Broken: version: 3.1.2, branch: addon_v3.0_octane2021 (modified), commit date: 2022-04-27 03:29, hash: rB5dfefe58f084
Worked: (newest version of Blender that worked as expected)

Short description of error
If you have weight paint that is being set dynamically - for example by using dynamic paint and a shape key that uses vertex group same as the one being modified, there is no change in the shape even if weight was changed.

Exact steps for others to reproduce the error
Reproduction steps:

1. Create sphere
2. Create Basis and Key 1 shape keys and set value to 1
3. Enter edit mode and modify the shape key (for example resize the sphere), then exit edit mode
4. Add Vertex group
5. Set the new group in the Vertex Group in the shape key
6. Add dynamic paint, canvas, Surface Type : Weight, Output: Vertex Group : Group
7. Add second sphere
8. On second sphere add Dynamic Paint : Brush, surface: Volume + Proximity
9. Create animation where one ball is 2 ball lengths apart and gets as close as to touch the other ball

Result: In weight paint mode observe that weight changes, but the shape is not affected.
Bonus round: Paint the weight paint manually and observe object changing

I have tested it on Blender 3.1 and Blender 3.0, both have this problem