Build docker images when PR changes them

[rachitnigam]

This is an annoying problem that we keep running into where the docker image we use can be out of date from the one a PR is creating. @UnsignedByte added a hack in the Filament repo to check if the Dockerfile has changed and rebuild it if needed. We should copy that code into this repo as well.

[sampsyo]

That would help a bit! One slightly "higher-coverage" option (that would not require explicitly hashing the Dockerfile) would be to just condition the Docker build on changes to certain paths directly. For example, this setup for the Bril repo only tries to build the docs if the docs have changed:
https://github.com/sampsyo/bril/blob/34133101a68bb50ae0fc8083857a3e3bd6bae260/.github/workflows/docs.yaml#L4-L9

We could do the same thing for the Docker image, conditioning it on Dockerfile and anything else that we think needs to bump the image (Cargo.lock?).