How to detect if a humanoid is climbing for 3 seconds continuously?

In your code snippet you aren't checking if the player is still climbing after the 3 seconds have passed. You can get the humanoid's current state with the GetState method. Just checking if the state is still "Climbing" may not be enough for you as the player might've started climbing again during that 3 seconds. To avoid this, you can listen to state changes and cancel the procedure if the state changed before the 3 seconds passed.

local player = game.Players.LocalPlayer
local character = player.Character or player.CharacterAdded:Wait()
local humanoid = character:WaitForChild("Humanoid")

local function flingCharacter(character, force)
	force = force or -50
	character.PrimaryPart.Velocity = character.PrimaryPart.CFrame.lookVector * force
end

humanoid.Climbing:Connect(function()
	--	Listen for changes in the humanoid's state
	local stateChangedConnection 
	stateChangedConnection = humanoid.StateChanged:Connect(function(_, newState)
		--	If the player jumped, then drop them from the ladder
		if newState == Enum.HumanoidStateType.Jumping then
			flingCharacter(character)
		end
		stateChangedConnection:Disconnect()
	end)
	
	wait(3)
	
	--	If the state hasn't changed, we can assume they were climbing the whole time
	if stateChangedConnection.connected then
		stateChangedConnection:Disconnect()
		flingCharacter(character)
	end
end)

local character = player.Character or player.CharacterAdded:Wait() This line of code is there in case the character existed before the script was executed. It assigns the current character of the player to the "character" variable and if it's not found, it waits until the character is added using the Wait method of RBXScriptSignal. The CharacterAdded event's only parameter is the new character, which is then assigned to the "character" variable as the Wait method returns it.