My sink is on one side of my galley kitchen and the fridge is on the other. The living room doorway is between the water line to the sink and the fridge. The house sits on a slab so there is no basement.
Can I run a line over the doorway?
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Sure, the water is under pressure, it'll get there. As far as static pressure loss, the entrance and exit are all that matter, the in between makes no difference. Friction loss is an issue with long distances, but ice makers and dispensers have very low demand, so unless your house pressure is very low already, it shouldn't be an issue. The only real drawback would be the appearance if you have no convenient ways to disguise the run.
I actually don't think that would be a good idea. Typically a header exists above a doorway and cripple studs on top of that. You couldn't drill through the header and I wouldn't run any type of line directly over a door like that.
If it were me I would drill a hole in the wall between the studs and fish it up to the ceiling and run it over to where it can be dropped where the fridge goes, but then my home has PEX plumbing so that of course would be possible.