How to tween a camera FOV?

The problem is that you're not playing the Tween. A Tween object has a Play() function which does what you think; it plays the tween.

However, to accomplish this we will need to reference the tween somehow, so we just save it as a local variable.

-- Services
local tweenService = game:GetService("TweenService")

-- Variables
local userInput = game:GetService("UserInputService")
local cam = workspace.CurrentCamera

userInput.InputBegan:Connect(function(input)
    if input.KeyCode == Enum.KeyCode.LeftShift then
        local t = tweenService:Create(cam, TweenInfo.new(0.2, Enum.EasingStyle.Sine, Enum.EasingDirection.InOut), {FieldOfView = 80})
	t:Play()
        script.Parent.Humanoid.WalkSpeed = 30
    end
end)

userInput.InputEnded:Connect(function(input)
    if input.KeyCode == Enum.KeyCode.LeftShift then
        local t = tweenService:Create(cam, TweenInfo.new(0.2, Enum.EasingStyle.Sine, Enum.EasingDirection.InOut), {FieldOfView = 70})
	t:Play()
        script.Parent.Humanoid.WalkSpeed = 16
    end
end)

Even with this, there's still room for improvement. If a player were to stop running shortly after having started running it will look a bit weird. So to fix this you simply use the Cancel() function of the Tween object to reset the animation.

I used some variables so that each tween is "Saved" but also readable by the entire script, thus solving our problem.

-- Services
local tweenService = game:GetService("TweenService")

-- Variables
local userInput = game:GetService("UserInputService")
local cam = workspace.CurrentCamera
local t0 -- New variable
local t1 -- Also new variable

userInput.InputBegan:Connect(function(input)
    if input.KeyCode == Enum.KeyCode.LeftShift then
        t0 = tweenService:Create(cam, TweenInfo.new(0.2, Enum.EasingStyle.Sine, Enum.EasingDirection.InOut), {FieldOfView = 80})
	t0:Play()
	if t1 then -- Make sure that the variable is set
		t1:Cancel()
	end
        script.Parent.Humanoid.WalkSpeed = 30
    end
end)

userInput.InputEnded:Connect(function(input)
    if input.KeyCode == Enum.KeyCode.LeftShift then
        t1 = tweenService:Create(cam, TweenInfo.new(0.2, Enum.EasingStyle.Sine, Enum.EasingDirection.InOut), {FieldOfView = 70})
	t1:Play()
	t0:Cancel() -- We don't need an if statement here, since t0 will be defined before it needs to be called.
        script.Parent.Humanoid.WalkSpeed = 16
    end
end)