First:

Your for loops are nested -- for each tiny step of the the left door at the very beginning, everything else is done also.

You don't want tons of ends stacked after each other like that, you want each for loop to end immediately before the next one starts.

You don't have any waits in this script, so everything will happen all at once. If you want to see animation, you need there to be pauses.

Second: You have so much repetition. Use a function! Something simple like this:

function moveY(part, fromY, toY)
    for y = fromY, toY, 0.5 then
        part.Position = Vector3.new(0, y, 0)
    end
end

However, that's probably not what you want. That will also move it horizontally, since you're saying you want the x and z to both be 0. Position is also usually not what you want; you want to set CFrame.

-- Let's just say how much it moves up, rather than from/to.
-- You can just leave in the difference when you call it, e.g.,
-- moveY(LeftDoor, 88.5 - 88.3)
-- in the first time this is used.
function moveY(part, difference)
    local up = 0.5
    for y = 0, difference, up do
        part.CFrame = part.CFrame + Vector3.new(0, up, 0)
        wait() -- Necessary to *see* this!
    end
end

Another problem is that this can only go up -- if the difference is negative, the for loop will just not happen. We need to fix that. Also, we should probably let a different speed for moving up or down be given (optionally):

function moveY(part, difference, by)
    local up
    if difference < 0 then
        up = -(by or 0.5)
    else
        up = (by or 0.5)
    end
    for y = 0, difference, up do
        part.CFrame = part.CFrame + Vector3.new(0, up, 0)
        wait()
    end
end

The definition of up can be more succinct using the ternary operator:

    local up = (difference < 0 and -1 or 1) * (by or 0.5)

Using this moveY and equivalent moveX and moveZ functions, we can get much, much shorter code:

-- this:
        for i = 88.3, 88.5, .2 do
            --Moves the position up by .2 each time
            LeftDoor.Position = Vector3.new(0,i,0)
        end
-- is replaced by this:
        moveY(LeftDoor, 88.5 - 88.3, 0.2)

We also have to make an equivalent function for rotations:

function spinX(part, angle, by)
    local spin = (difference < 0 and -1 or 1) * (by or 1) -- degrees
    for t = 0, angle, spin do
        part.CFrame = part.CFrame * CFrame.Angles( math.rad( spin ), 0, 0)
        wait()
    end
end

The resulting code (for the On function anyway) is much, much clearer and simpler:

function On()
    while "script.Parent.ActivateButton.On.Value == false" do
        spinX(LeftDoor, 12 - 0, 1)
        moveY(LeftDoor, 88.5 - 88.3, 0.2)
        spinX(LeftDoor, 18 - 12, 1)
        moveY(LeftDoor, 88.7 - 88.5, 0.2)
        spinX(LeftDoor, 25 - 18, 1)
        moveZ(LeftDoor, -204.4 - -204.5, .1)
        moveY(LeftDoor, 88.8 - 88.7, 0.1)
        spinX(LeftDoor, 28 - 25, 1)
        moveZ(LeftDoor, -204.35 - -204.4, .05)
        spinX(LeftDoor, 30 - 28, 1)
        moveY(LeftDoor, 88.9 - 88.7, 0.1)
        spinX(LeftDoor, 33 - 30, 1)
        moveY(LeftDoor, 89 - 88.9, .1)
        moveZ(LeftDoor, -204.3 - -204.35, .05)
        spinX(LeftDoor, 37 - 33, 1)
        moveY(LeftDoor, 89.1 - 89, .1)
        moveZ(LeftDoor, -204.25 - -204.3, .05)
        spinX(LeftDoor, 41 - 37, 1)
        moveY(LeftDoor, 89.2 - 89.1, 0.1)
        moveZ(LeftDoor, -204.2 - -204.25, .05)
        spinX(LeftDoor, 45 - 41, 1)
        moveZ(LeftDoor, -204.1 - -204.2, .05)
        moveY(LeftDoor, 89.3 - 89.2, 0.1)
        spinX(LeftDoor, 50 - 45, 1)
        moveZ(LeftDoor, -204.05 - -204.2, .05)
        spinX(LeftDoor, 53 - 50, 1)
        moveY(LeftDoor, 89.4 - 89.3, .1)
        moveZ(LeftDoor, -204 - -204.2, .05)
        spinX(LeftDoor, 56 - 53, 1)
        moveZ(LeftDoor, -203.95 - -204, .05)
        spinX(LeftDoor, 59 - 56, 1)
        moveY(LeftDoor, 89.5 - 89.4, .1)
        moveZ(LeftDoor, -203.9 - -203.95, .05)
        spinX(LeftDoor, 62 - 59, 1)
        moveZ(LeftDoor, -203.85 - -203.9, .05)
        spinX(LeftDoor, 65 - 62, 1)
        moveZ(LeftDoor, -203.75 - -203.85, .05)
        spinX(LeftDoor, 67 - 65, 1)
        moveZ(LeftDoor, -203.7 - -203.75, .05)
        spinX(LeftDoor, 68 - 67, 1)
        moveY(LeftDoor, 89.55 - 89.4, .1)
        moveZ(LeftDoor, -203.65 - -203.7, .05)
        spinX(LeftDoor, 70 - 68, 1)
        moveZ(LeftDoor, -203.6 - -203.65, .05)
        spinX(LeftDoor, 72 - 70, 1)
        moveZ(LeftDoor, -203.55 - -203.6, .05)
        spinX(LeftDoor, 74 - 72, 1)
        moveZ(LeftDoor, -203.5 - -203.55, .05)
        spinX(LeftDoor, 76 - 74, 1)
        moveZ(LeftDoor, -203.45 - -203.55, .05)
        spinX(LeftDoor, 78 - 76, 1)
        moveZ(LeftDoor, -203.4 - -203.45, .05)
        spinX(LeftDoor, 79 - 78, 1)
        moveZ(LeftDoor, -203.35 - -203.4, .05)
        spinX(LeftDoor, 81 - 79, 1)
        moveZ(LeftDoor, -203.3 - -203.35, .05)
        spinX(LeftDoor, 83 - 81, 1)
        moveZ(LeftDoor, -203.25 - -203.3, .05)
        spinX(LeftDoor, 86 - 83, 1)
        moveZ(LeftDoor, -203.2 - -203.25, .05)
        spinX(LeftDoor, 88 - 86, 1)
        moveZ(LeftDoor, -203.15 - -203.2, .05)
        spinX(LeftDoor, 90 - 88, 1)
        moveZ(LeftDoor, -203.1 - -203.15, .05)
    end
end

I cannot fathom how you determined these numbers, however.

In addition, the while loop is wrong. Quotes mean text. Lua ignores that and just says, "okay, you're giving me text, I'm happy about that! Let's loop forever." Remove the quotes for it to make sense.

Lessons Learned

Don't write huge amounts of code! If it's a struggle, you're probably doing something wrong.

While the code here is immensely better, it could still be significantly improved.