Can TYPO3 run without local file storage for the typo3temp directory?
TYPO3 instances need some work on your side for being scaled horizontally.
I am assuming TYPO3 v9+ and a default composer based installation.
You need to share at least public/typo3temp, public/fileadmin and ./var between instances. That is because their state is very much tied to the database state. See https://docs.typo3.org/m/typo3/reference-coreapi/master/en-us/ApiOverview/DirectoryStructure/Index.html.
The rest is part of your TYPO3 deployment and could be baked into container images. I am pretty sure for live deployment none of these need to be writable. You might lose some admin backend functionality like through-the-backend configuration changes but that should be obvious.
Raising site performance
That said, getting more performance out of TYPO3 is a lot easier with a reverse caching proxy in front of it than by multiplying PHP instances. There is also https://extensions.typo3.org/extension/staticfilecache/ which generates static HTML for cachable content and has a good reputation, too.