For longevity of the truck, I would turn it off if you don’t need it.  The tow/haul mode in that Tundra holds lower gears longer when you accelerate or slow down (more stress to transmission), and also adds engine braking to help slow the vehicle down. It is better to use the truck’s brakes to slow down and not rely on the engine - can wear it out, and brakes are cheaper than a new engine.

I'm going to guess you want to leave it in tow haul mode for "sporty" driving.

That's a bad idea. Increased wear on the entire drivetrain, poor MPG....the truck is not a race car. Don't expect to be able to hammer on it everywhere without repercussion.

Why would you?