How to allow use of a shift-sprint script when Game Pass owned?

You can place the script in somewhere like ServerStorage and then run it only on the players who have purchased the gamepass once they join, by cloning it and parenting it to their PlayerGui.

Server script:

local gamePassId = 123
local theScript = game:GetService("ServerStorage").theScript

game:GetService("Players").PlayerAdded:Connect(function(plr)
	if game:GetService("MarketplaceService"):UserOwnsGamePassAsync(plr.UserId, gamePassId) then
		theScript:Clone().Parent = plr:WaitForChild("PlayerGui")
	end
end)

Btw you should use UserInputService.InputBegan instead of Mouse.KeyDown.

The localscript:

local plr = game:GetService("Players").LocalPlayer

script.Parent = plr:WaitForChild("PlayerScripts") --so it keeps running after respawn

game:GetService("UserInputService").InputBegan:Connect(function(iobj, gp)
	if gp then return end
	if iobj.KeyCode == Enum.KeyCode.LeftShift and plr.Character then
		plr.Character:WaitForChild("Humanoid").WalkSpeed = 50
	end
end)

game:GetService("UserInputService").InputEnded:Connect(function(iobj, gp)
	if gp then return end
	if iobj.KeyCode == Enum.KeyCode.LeftShift and plr.Character then
		plr.Character:WaitForChild("Humanoid").WalkSpeed = 16
	end
end)