Obviously one cost is time. Lots of loading-screen-like transitions will force the player to spend a lot of time waiting around doing nothing, and probably getting bored. I can't tell you how annoyed I've been when I've had to exit/enter a building in a city in Skyrim twice in a row, spending nearly five minutes just because I forgot to leave some random item. Something that should have taken me a minute tops, suddenly took about five times longer.
It's also quite immersion breaking if the player finds a way to overcome the invisible walls between zones, and find themselves in limbo. Not to mention that depending on how your gameplay works, it may get the player stuck forever in limbo, ruining all progress (this happened to me in Arkham City after having beat the whole campaign, thankfully, but still was annoying)
I've also disliked how some games allow you to be in a position to affect another area (say, on top of a wall above a battlefield), but you couldn't affect the other area simply because it was 'another area' and you had to go load into it before you could affect it.
There's other issues too if your game is multiplayer. For example, in EVE and its open pvp nature, having specific entrance/exit points in a system causes them to act as choke points for player killings, traps, etc. In EVE its more or less a game mechanic they've worked in, but it's something I've always hated.
The only time I've really seen player feedback for a game is in GW2. That game is very much divided into areas - and quite the opposite of what most games in its genre are doing. I've only seen a few people complaint about the area transitions, but most were ok with it, especially given that the areas are quite large. I imagine if the areas were smaller, it would be a lot more annoying to the player. Well, I've also seen some random feedback from people regarding the next gen consoles, and how they expect no loading areas. Not sure if it's a make-or-break feature, but it can hurt your game if done horribly wrong.
There's also technical difficulties involved in making area transitions. The most obvious one I can think of is AI. If you have any sort of AI, friendly or otherwise, and they have to move at some point - follow/attack the player - you must either write the AI with the ability to use the area transitions, or to give the player a possible exploit if the AI doesn't follow through area transitions. I've seen some pretty bad issues in TES games with AI using area transitions. They get stuck right before exiting, I have to re-enter and push them, or they enter an area before me, and when I enter only a second later, I find they've seemingly disappeared, having advanced much further than the second difference between them using the door and me using the door would allow.
I think as long as you make the areas large enough, minimize the loading times. make the separations as natural as possible, and don't divide areas that look like they should logistically be the same area, the area transitions aren't a major problem. To each his own, but if any of those things above stick out in a game, I often start disliking the area transitions.