No, it's done all the time, you just need to use the right cable.
Depending on -where- you run the cable, you will need a specific type.
The standard outdoor cable usually isn't suitable for in-buildng use due to flame propagation and smoke issues. If there was a fire in the building, your new cable could be a source of issues. The cable itself can burn and help spread the fire. The burning cable on it's own can introduce all kinds of toxic gases into the atmosphere.
Where you run the cable is key:
If it's running between floors, you will need to use (at minimum) a riser rated cable. These will usually be marked with "CMR" on the jacket somewhere. This means its a communications cable, riser rated.
If the cable is run in any air handling spaces, such as in a drop ceiling that is used as part of the air return system, then you will need to use a plenum rated cable. These will be marked with CMP on the jacket.
These riser and plenum rated cables will usually be different than what you run outdoors. They'll likely cost more, too.
At absolute minimum, I'd recommend running a riser rated coax cable. You can find this fairly easily in the RG-6 rated cables as they are often used for cable TV distribution inside buildings. Finding Plenum rated will be a bit harder, as in not likely at your local hardware store, but it's out there.