This feature was added for JavaScript and TypeScript in VS Code 1.24 (tracking issue)
When you move or rename a file, you will now be prompted to see if you want to update imports:

This is controlled by the javascript.updateImportsOnFileMove.enabled and typescript.updateImportsOnFileMove.enabled settings. Valid values are:
"prompt"— The default. Prompt for each file rename/move"always"— Always try to update imports automatically without prompting"never"— Do not update imports and do not prompt