DevCameraOcclusionMode, Online vs Offline issues?

To prevent the camera from getting stuck, we can wait before executing the script:

repeat wait() until game.Players.LocalPlayer;
repeat wait() until game.Players.LocalPlayer.Character;

Also, we can make the camera subject the Humanoid, rather than the LocalPlayer.

camera.CameraSubject = plr.Character.Humanoid

Also, put the part inside the plr.Character so it gets destroyed when the player dies.

Final Script:

repeat wait() until game.Players.LocalPlayer;
repeat wait() until game.Players.LocalPlayer.Character;

local plr = game.Players.LocalPlayer
local chr = plr.Character or plr.CharacterAdded:wait()

local root = chr:WaitForChild("HumanoidRootPart")
local leftLeg = chr:WaitForChild("Left Leg")
local rightLeg = chr:WaitForChild("Right Leg")
local hum = chr:FindFirstChild("Humanoid")

local camera = workspace.CurrentCamera
camera = workspace.CurrentCamera
camera.CameraType = Enum.CameraType.Scriptable;
camera.CameraSubject = plr.Character.Humanoid;


local part = Instance.new("Part",plr.Character)
part.CanCollide = false;
part.Anchored = true;
part.FormFactor = Enum.FormFactor.Custom
part.Size = Vector3.new(7,.1,6)
part.Transparency = 1;
local gui = Instance.new("SurfaceGui",part)
gui.Face = "Top"
local decal = Instance.new("ImageLabel",gui)
decal.Size = UDim2.new(1,60,.8,0)
decal.Position = UDim2.new(.1,-20,.2,-15)
decal.Image = "http://www.roblox.com/asset/?id=276509139"
decal.BackgroundTransparency = 1;
local falling = false;

local mouse = plr:GetMouse()
mouse.TargetFilter = workspace.Map

function getBase(side)
    local sign = (side == "up" and 1 or -1);
    local ray = Ray.new(root.Position, Vector3.new(0, 1000 * sign, 0))
    local hit, position = workspace:FindPartOnRayWithIgnoreList(ray, {chr,part,camera})
    return hit,position;
end

--[[
hum.Died:connect(function()
    part:Destroy()
    chr = plr.CharacterAdded:wait()
    root = chr:WaitForChild("HumanoidRootPart")
    leftLeg = chr:WaitForChild("Left Leg")
    rightLeg = chr:WaitForChild("Right Leg")
    hum = chr:FindFirstChild("Humanoid")
end)]]

hum.FreeFalling:connect(function(plrFalling)
falling = plrFalling
end)

game:GetService("RunService").RenderStepped:connect(function()
camera.CoordinateFrame = CFrame.new(CFrame.new(root.Position + Vector3.new(-20,25,0)).p,root.Position)
root.CFrame = CFrame.new(root.Position, Vector3.new(mouse.Hit.p.X, root.Position.Y, mouse.Hit.p.Z))

if falling then
    local parter,pos = getBase("down");
    part.CFrame = CFrame.new(pos, pos + Vector3.new(root.CFrame.lookVector.X, 0, root.CFrame.lookVector.Z))  --[[*CFrame.Angles(0,math.rad(-90),0);]]
elseif hum.Jump == false then
    part.CFrame = (root.CFrame - (Vector3.new(0,leftLeg.Size.Y,0) + Vector3.new(0,root.Size.Y/2,0)))  * CFrame.new(-.5,0,0);
end
end)

To fix your DevCameraOcclusionMode problem, I modified the built in camera script to automatically go into Invisicam mode. Since there are modules and stuff inside it, I will just link you my model.

However if you want to change it yourself, these are the changes I made to the script:

--Around line 150
    --if newOcclusionMode == Enum.DevCameraOcclusionMode.Zoom then
    --  EnabledOcclusion = PopperCam
    --elseif newOcclusionMode == Enum.DevCameraOcclusionMode.Invisicam then
        EnabledOcclusion = Invisicam -- Set it to Invisicam mode
    --else
    --  EnabledOcclusion = false
    --end

Then at the beginning I added this to take out the CameraScript roblox puts in.

repeat wait() until game.Players.LocalPlayer;
for a,v in pairs(script.Parent:GetChildren()) do
    if v.Name == "CameraScript" and v ~= script then
        v:Destroy()
    end
end