Read named pipe once every time input is written

In your case, you can do simply:

tail -f $pipe | sh & 

no need for loops.

One way to do it:

#! /usr/bin/env bash
pipe=/path/to/pipe
[ -p "$pipe" ] || mkfifo -m 0600 "$pipe" || exit 1
while :; do
    while read -r cmd; do
        if [ "$cmd" ]; then
            printf 'Running %s ...\n' "$cmd"
            # sh -c "$cmd" sh
        fi
    done <"$pipe"
done

It's probably a really bad idea to uncomment the sh line though.