With the Space X success at re-using rockets, my thought is that the second stage could be refueled at the ISS. Similarly, fuel could be sent to orbit Mars. My thought is it would allow for greater acceleration and therefore cut travel time.
SpaceX's intent is to use orbital refueling to allow Starship to reach destinations beyond low Earth orbit. However, these plans simply involve meeting up directly with the tanker to transfer propellant.
The ISS has no facilities for storing or transferring the liquid methane and oxygen Starship uses as propellant. If it did, you'd still need two separate rendezvous and docking operations when directly meeting with the tanker only requires one. And finally, the ISS is in a fairly highly inclined orbit. Not only would this limit the payload the Starships could carry, it wouldn't necessarily line up with your intended destination.
The point from the Earth orbit from where we launch has little influence on the travel time. Currently all missions to Mars use the less energetic path. Basically, we accelerate the spacecraft until we reach an elliptical transfer orbit around the Sun with the periapsis (closes point to the sun) in Earth Orbit and the apoapsis (further point from the sun) intersecting Mars orbit. Once in that that trajectory the spacecraft coasts the rest of the way, braking once it reaches Mars's sphere of influence. This is the best you can get with current technologies and takes around 259 days, about 8.5 month, and most missions have barely enough fuel to perform this maneuvers, most Martian Rovers don't even use fuel to break, instead perform an aerobreak maneuver in Mars atmosphere. The only option to cut that time is to accelerate during extended periods of time during the first half of the trip and break during the same amount of time during the second half. We can't do that yet, and definitely not with chemical rockets, although some ion and plasma engines (some already in prototype and demonstration stage) have the theoretical capacity to reduce the trip to 2 months. Starship is designed to use the first approach and it will be limited to the less energetic path, so refueling in the ISS or refueling using a tanker makes little to no difference to the travel time, is completely limited to that specific transfer orbit. Added to this, the ISS has no infrastructure for refueling or storage and given the complexity of the structure it would be incredibly expensive to implement something like that, specially compared to use a reusable tanker from SpaceX.
Mars. Would it save time to launch from the ISS?
My understanding is it would not. The ISS is an international collaboration, and it's mission is as much political as it is scientific. Russia (and it's predecessor political entity the USSR) is a major partner in this program and they had their own input on how the program would operate. One requirement that they had was an orbit in which they could reach with minimal fuel expense from their space ports. Russia is a very northerly nation, so far north that most or all of it's seaports freeze over in the winter (which has lead to a number of land wars in Asia, but I digress). So far north that it is difficult to reach an orbit close enough to the orbital plane to get any advantage as a launching point for interplanetary travel.
The original intent of what was to become the ISS was as a stopping off point for missions beyond Earth. But to get the Russians/Soviets to agree to this required an orbit that could be reached from their spaceports with the rockets they had that were rated for carrying humans. The orbit was changed and the ISS was born.
With the ISS so far from the orbital plane it would take more fuel to correct for this high inclination orbit. It would take less fuel to not stop at the ISS, or to launch a new space station as a refueling stop.
The ISS is getting very old at this point, and there's many nations that would prefer to put their limited budgets in manned space flight into a program that would allow manned missions to other planets. Talk of abandoning the ISS keeps coming up, but perhaps with the Russians continuing with the remains of ISS after everyone else leaves. This might involve the parts owned by other nations being donated or sold to Russia, the parts somehow moved to another orbit, or perhaps most likely separated off and dropped into the atmosphere to burn up.
It is so expensive to launch from the ISS to Mars, or any other destination in the solar system, that it would be cheaper to drop the ISS in the ocean and start over with a new space station.
I'm going from memory here based on my amateur knowledge of orbital mechanics so if someone wants to jump in with some real world numbers then I'd like to see it.