I've died quite a few times, does it have any impact on the game as a whole, or is it just a question of restarting that particular scenario?
Mass-effect – Is dying a bad thing
mass-effect-3
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The following is the memorial plaque wall when you start with a new character and select the "numerous" option. As you'll note, most of the squad members from Mass Effect 2 are still alive:
Those names underlined in blue are the original crew members of the Normandy SR-1 who died in the beginning of Mass Effect 2, along with the two squad members Ashley Williams and Urdnot Wrex (marked by blue squares). Kaidan was chosen to be alive because I was playing a female Shepard; when playing a male Shepard, Ashley will be the Vermire survivour.
Those names underlined in orange are all crew and squad members from the Normandy SR-2. Those marked with an orange circle, Gabriella Daniels, Kenneth Donnelly, and Kelly Chambers, would have all made notable appearances in Mass Effect 3 had they not died. More specifically, the former two can be brought back aboard the Normandy in their usual positions as engineers, and Kelly Chambers is a potential romance/fish keeper* depending on your interactions with her in the previous game.
Thane Krios and Jacqueline Nought (Jack) appear to be the only squad member from Mass Effect 2 that's listed as having died, and are highlighted on the wall by the orange square.
Everyone else not on this list is presumed to be alive, and not one of the "numerous who died." In particular, Garrus, Tali, Miranda, and Jacob are all listed as alive. However, the game will act as though you never met Samara/Mordinth, Grunt, Katsumi, and Zaeed, effectively killing them. It will also act as though the original Council died during the attack on the Citadel in Mass Effect 1.
*that is, she returns your fish from Mass Effect 2 to you; you need the aquarium VI or extreme patience to keep them alive
The problem of course is that this is the same list of people who would have died if you had simply chosen the Ashley/Kaidan option. In comparing the two save files, one from a "numerous" selection, and one from a "Kaidan" selection (using male Shepard to keep a consistent Vermire casualty), there's actually very few differences between the two scenarios.
The tables below illustrate the difference in game variables before you perform any action:
PlotID | Numerous | Kaidan dies
--------+----------+------------
17807 | true | false
18005 | false | true
18122 | false | true
21569 | false | true
PlotInt | Numerous | Kaidan dies
--------+----------+------------
10393 | (unset) | 3
At the moment it's unfortunately unclear what these changes represent, because they don't directly map to any of the significant and/or labeled plot events that the game has. Presumably they have some noticeable impact, however, so I'm going to keep digging until I get to the bottom of this.
The dialog options you pick can have a large impact on the game - in many cases small choices can have larger effects further down the line. In fact, your choices in earlier Mass Effect games can and will come back to haunt you - even though it's been years since you picked them.
The clearest of these is the "Paragon" / "Renegade" system, which usually is up-right and down-right on the conversation wheel. Usually the "Paragon" option is the "boy scout/goody two shoes" option, and "Renegade" is the "punch stuff and listen to music your parents don't approve of" options.
Picking one of these two options will usually reward you with either Paragon or Renegade points. This, alongside your reputation (which is similar, but neutral instead of good or bad) can make it easier to influence people at certain key decision points.
There are many decision points in the game though that aren't clearly called out as such. Generally speaking you're going to want to get all the information you can before you make a choice in a conversation, because sometimes people will live or die based on these actions.
It's also a very good idea to consult with all of your crewmates between missions, as between almost every mission you've got a chance to talk with them, learn something new about them, and potentially influence them. Some missions and powers are only available if you have a close relationship with your squadmates.
It can be a staggering burden to always have the fate of the galaxy in your hands, so you might want to plan on making many saves, or plan on making several runthroughs of the game to experience all the different things that can happen. Don't feel paralyzed though - it's hard to paint yourself completely into a corner, although it is possible to lock yourself out of the "best ending" if you're not careful.
Action mode takes away your ability to choose, but it does not take away the effects of these decisions. You'll be steered along a particular course that has been laid out for the "action experience." This course seems to be a mix of Paragon and Renegade choices although I don't know if it's been exhaustively tested to date.
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Best Answer
In general, the only time dying has permanent consequence on the game (other than the obvious fact you just died) is during events that have been scripted to have an effect.
Examples include Shepard dying at the end of Mass Effect 2 if things went horribly, horribly wrong, during the "survive 5 waves" event of the Arrival DLC, the non-standard game over if you don't escape the Reapers during scanning in ME3, etc.
Otherwise, the game does not keep track of your deaths in a way that would have a meaningful impact on the game. In general, avoiding death is in your best interest for obvious reasons (quicksave often!), and if your death is going to have a meaningful impact on the story, like during a scripted event, you will notice it.