Users like to be control
Even though your data is always up to date, users like the perception that they can control the data stream or information refresh rate and hence I would recommend keeping the Pull to refresh so that users can exercise that action (even though it might not pull in new data). To quote this article from UX booth
The human need for control can be traced back to our earliest roots.
In psychologist Abraham Maslow’s well known hierarchy of needs, he
identifies our most basic needs as the physiological ones: health,
food, and sleep. All of these require a significant amount of
control—control over our environment in order to gather food, and
control over our own choices to avoid disease.
It’s therefore unsurprising that we seek out control in everything
from our relationships—everyone has “put their foot down” at some
point or another—to the color of our living room walls—even though the
old color was fine, it wasn’t perfect. It’s ingrained in us, and it
gives us a sense of comfort and well being.
This sense of control is very closely linked to what psychologists
call an “internal locus of control,” or the belief that our actions
have the power to impact and change a given situation. (Conversely, an
“external locus of control” is the belief that we are at the mercy of
external sources.) People with an internal locus of control are more
likely to be confident, take better care of themselves, and have lower
levels of stress.
As UX designers, we try to ensure users will have positive experiences
when using our products or accessing our services. This often
translates into empowering users, thus giving them the tools to find
their internal locus of control
Since you also want the user to be informed about when the data was last refreshed (either by a manual refresh or a dynamic refresh), I would also suggest showing them information inline on the last refresh state as shown below