One of the simplest still-lifes is the "beehive":
. # # . # . . # . # # .
If you remove the cell at one end, it will eat itself over the next few generations and nothing will remain.
I suspect there is a position with the properties that (1) its population remains nonzero but bounded, (2) if you change one cell you can make it go extinct, and (3) if you change one cell you can make it grow without limit (via glider guns or the like), but that would be much more difficult to construct.
There is this configuration which is stable:
.#. #.# .#.
If you take out any one of them, it will die.
If you start with four square blocks, this is stable.
# # . # # # # . # # . . . . . # # . # # # # . # #
On removing any one of the 4 innermost cells it will mutate and form another stable shape in 6 generations.
Remove any or all of the 4 corner cells and it will immediately self heal.
But if you remove any one of the other 8 outer cells it will mutate and die out in 33 generations.
I think this is actually a solution for your question as asked:
No cells live at all. It's a stable formation (of zero cells), and changing any cell (toggling any single cell to Live anywhere on the grid) creates a population that will indeed become extinct, in one generation.
specifically on a 3x3 grid. It's stable, but toggling any cell causes extinction.
(This pattern is the same as rhsquared's, but confining it to a 3x3 grid means it's a solution when removing any live cell or when adding any new live cell.)
Perhaps with some changes
to the question to clarify what is meant by an initial "position" an uninteresting answer like this one will be ruled out. :)