Can I play Mass Effect 3 in First Person Mode?

Tweaking the Field of View

1. Back up your Coalesced.bin and save files
   %ProgramFiles(x86)%\Origin Games\Mass Effect 3\BIOGame\CookedPCConsole\ %Username%\My Documents\Bioware\Mass Effect 3\Save\

2. Download the coalesced editor.

3. Run the editor, open Coalesced.bin

4. Navigate to
   bioinput.ini > sfxgame > sfxgamemodebase > bindings (multiple)
   Replace "Shared_Aim" line with
   ( Name="Shared_Aim", Command="SwapWeaponIfEmpty | TightAim | FOV 0 | OnRelease FOV 101 | OnRelease StopTightAim" )

5. Navigate to
   bioinput.ini > sfxgame > sfxgamemodecinematic > bindings (multiple)
   Add a line
   ( Name="RightMouseButton", Command="FOV 0")

6. Navigate to
   bioinput.ini > sfxgame > sfxgamemodeconversation > bindings (multiple)
   Add a line
   ( Name="RightMouseButton", Command="FOV 0")

7. Save, then close the editor.

Step 4 makes it so when you aim down the sights you use the normal zoomed fov, but go to a wider 101 degrees when you release the aim key. Feel free to change the value to suit.
Steps 5 and 6 allow you to reset the fov to default when a conversation or cinematic starts by pressing the Right Mouse Button, so as to avoid seeing visual glitches that would otherwise be hidden.

No, Mass Effect 3 is a third person shooter. Except for situations where you're zooming with a scoped weapon, you're always going to be in third person.

When aiming down sights on other weapons, the camera pulls in slightly, but otherwise there's really very little in the way of camera controls.