I'm wondering if EFS on AWS is fast enough to store PHP and execute PHP code? Or does the time it takes to access the files on the network when an HTTP request comes in, make this a poor solution?
It all depends on your use case. Is it slower than EBS or ephemeral storage? Yes. Is it too slow for your use case? Hard to say.
When using EFS you are bound to the network IO availability on the shared infrastructure that your EC2 VM is running on.