(Truth in advertising... I don't know much about this ;-) )

Reading the documentation where it says

On systems that support pipes, OpenRead["!command"] runs the external program specified by command, and opens a pipe to get input from it.

perhaps that suggests that you need to open the pipe directly in the argument of OpenRead rather than first creating it using Run?

When I try this it seems to work:

In[3]:= OpenRead["!mkfifo /tmp/test.fifo"]

Out[3]= InputStream["!mkfifo /tmp/test.fifo", 237]