I doubt a separate server will improve performance. Modern OSes are smart about keeping files cached in RAM after they are read. The first time you read a template file, it is cached and all subsequent accesses are fast. The OS is even smart enough to update the disk cache when the file is written. The latency of talking to a server process is much higher.

If the processing of the templates if complicated, there could be some advantage to caching the result. If you want a shared or persistent cache, storing it to disk makes the most sense. The disk cache will make read efficient and it is shared between all processes.