Does instant transportation effect game immersion?
Short answer: No.

Long answer: Instant travel only has an effect on immersion if travel time is a significant and worthwhile part of gameplay. The game's primary focal points are combat, crafting, and narrative (not necessarily in that order), which are affected by travel to varying degrees.

Travel affects combat (and gathering) the most, but only when you're in the field, where combat and gathering skills are actually relevant. Outside of that, travel is irrelevant, unless you're going to be mining iron or killing monsters on the airship.

For crafting, travel doesn't add anything. Having a 10-minute airship ride isn't going to give crafters anything they couldn't already get sitting in Ul'dah all day.

For narrative, travel is only relevant if the plot deems it so. If plot-relevant characters can be found on the airship, then having a scene on the airship prior to reaching your destination is an important break in the action, but only then; if there's no plot events occurring, having the extra scene in-between adds nothing to the experience.

Immersion is less about mimicking reality, and more about maintaining the audience's interest. People are more immersed when there's something going on that's keeping them active in the game than when nothing is going on and the only way to pass the time is to troll linkshell chat.