It seems to me that when people note videogame influence, they're usually noting "speed of play at the cost of verisimilitude."
Things like cheap resurrection magic. Makes the game flow nice, but how often did you wonder why you couldn't just cast
Life to raise Aeris from the dead? (Of course, it's usually 'fluffed' out of the way by calling the 'hp falls to 0' condition KO or Swoon or something non-deadly)
Things like quest cards. It organizes the thoughts, but it kind of puts a control on quests arising organically from the game.
Things like 'refreshing' abilities. It helps you go without having a five-minute dungeon expedition, but it also means your use of power doesn't tire or exhaust you, except for a few moments.
Most of the versimilitude issues can be fluffed out of the way, if they even need to be. Calling 0 hp "Unconcious" and not "dead," making resurrection more like snapping out of an action-movie karate-chop coma. Quest cards that pertain directly to things arising organically from the PC's actions. Refreshing abilities as bursts of adrenaline.
But still, when most people are looking for the videogame influence, this is what they're looking at, because it's what videogames are famous for -- fun play that doesn't care much about the ramifications of it's rules beyond the play itself. PacMan didn't have a narrative, the Mushroom Kingdom was an excuse to run and jump, and Final Fantasy has often been accuse of being an excersise in "pressing a button to advance the plot to the next FMV" more than a game. D&D has to concern itself with a continuous world, so a lot of people see it as a bad influence, at least in large amounts, if 'realism' is eroded for quickness of play. In addition, there's always those people who take a sort of warped pride in how long and how thorough they are about the detail that D&D allows.
In this resepct, D&D4 is 'more videogame' than previous editions, mostly because of the stated goal to streamline everything, and the fact that these streamlining things are being explained away, rather than rising organically from the campaign. I just don't necessarily buy that streamlining the game is NECESSARILY treating it more like a videogame, but it is a common evolutionary trend in both D&D and in videogames.
IMO, a game like FFZ shows it's true colors more boldly.
Death is permenant. Characters just don't actually 'die' that often. They might fall unconcious, become 'swooned,' get 'KO'd', or even get captured (only to wake up later), but true death doesn't come from an accident of the dice. It comes from the player's choice to sacrifice themselves (and give their party some great benefits, and themselves a shiny new character or taking over an NPC role).
The Mission system is set up so that players get to choose which quests and side quests to explore. There is an obvious narrative, but the players aren't shoehorned into it, and the GM can manuever it around a lot from session to session.
MP is restored with every encounter, and HP can be healed with MP. The challenges more lie in individual encounters, especially in the great speed with which your HP can drop and the need to get healing available in a reasonable casting time that is significant enough to stop it from dropping.
Characters rarely miss. If they do, it's usually because of a special feature or peice of equipment. Hitting is the assumed norm, and characters vary in the types and sizes of damage dealt rather than in missing more often.
These are all directly videogame-inspired rules that make the game much, much more fun. They do require a different kind of suspension of disbelief than, say, 1e D&D did. It's not really game-breaking, though it probably is for some who prefer a feeling of more helpless characters. Which is why FFZ isn't necessarily for everyone. But it obviously is for those who enjoy both 4e and Final Fantasy.
