Should I turn off image stabilization when I am taking a landscape shot?

Let me quote the manual:

When you use a tripod, the Image Stabilizer should be turned off to save battery power.

The manuals of both lenses (1) (2) don't mention any automatical turn-off feature but contain the above instruction, meaning you have to manually take care of this.

As you see, there are no detrimental effects on the photos when keeping the IS turned on with those two modern lenses. It just helps you to save energy when you turn it off in scenarios when it's not needed. Personally, I always carry along backup batteries and don't worry too much about the IS's power consumption.

Generally speaking Canon's manuals are really good, trustworthy and cover a lot of details and you can download all of them as far as I know. I always recommend to have a look at the manual first.

It is widely agreed upon to turn IS off on a tripod: The best thing that could happen, is that it does not improve the shot.

While most photographers experience a bit reduction in sharpness as the stabilizer may try to "wiggle" a bit around to calibrate and see if it can reduce shake. This is usually below the threshold of the shake we do while handholding the cam, so there is no visible effect. However on tripod it can become an issue.

There might also be an effect of a bit of a feedback loop.

Citation from Canon Europe Website regarding IS lenses:

When using certain early models of IS lenses with a tripod it was necessary to turn off the IS function. This is because of a phenomenon known as ‘Shake Return’. Shake Return occurs when the IS system tries to correct vibrations to which the system itself contributes. When the IS lens sits on a tripod, the IS detection gyros pick up any tiny vibrations or movement; these might be caused by the tripod being knocked, or the photographer adjusting a camera setting.

The IS system then swings into action to correct that movement. The movement of the IS lens group causes its own minute vibration, which is in turn detected by the movement sensor, which triggers another correction. This ‘feedback loop’ can continue endlessly, resulting in the addition of unwanted blur to images that would be sharper if the IS function was switched off.

As Matt already pointed out, most camera manuals even indicate turning it off on a tripod.

Some cameras or lenses may have tripod detection. Citation from Canon again:

The EF70-200mm f/2.8L IS USM incorporates the same automatic tripod detection function that is used with Canon’s super-telephoto IS lenses. Output signals from the vibration gyro are analyzed to determine if the lens is hand-held or mounted on a tripod. When the image stabilizer function is turned on and the lens detects that a tripod is in use, the shift lens is electronically centered and locked on the optical axis, preventing improper operation.

TLDR: At best IS does nothing on a tripod. In worst case, it ruins your sharpness. Switch it off to be on the safe side.

The correct answer totally depends upon the specific lens and also upon the manner in which you desire to use it.

When using a lens on a tripod, some lenses require you to turn IS off, some do it automatically for you, and others actually have IS modes specifically created for tripod use. The last category includes Canon's latest Super Telephoto series that are tuned to reduce the effects of mirror slap when mounted on a tripod.

Most of the problems with "feedback" that actually made lenses less stable by sensing the vibration of the IS motor and trying to correct it (which created more vibration that it tried to correct (which created more vibration...(etc.))) were with a handful of the very early first generation stabilized lenses produced two decades ago. But since one did actually need to turn IS off with those lenses to get best performance, there are still many photographers around today who preach a gospel that says, "IS must always be turned off when a camera and lens are mounted on a tripod." This is simply not true for most current lenses. Warnings written for those handful of early IS lenses that couldn't sense when they were on a tripod also persist in the user manuals of many lenses that do sense when they are tripod mounted.

Whether to use IS when the lens is attached to a tripod comes down to usage: If you're using shutter speeds slow enough to induce blur from camera movement, whether due to a less than rock solid tripod or by effects of wind when outside or by vibrations of a floor when inside or by your shooting technique (if you're still firing the shutter with the camera's button you're almost certainly moving it), then by all means turn it on and see what happens. As with many shooting techniques, this should probably be experimented with before a shoot that has to be done right the first time!

I tend to leave IS off when shooting from a monopod as long as the shutter speeds I'm using are at least twice the focal length with crop factor included in the calculation. If I'm using a Canon APS-C body with 1.6x crop factor and a 200mm lens, as long as my shutter speeds are above 1/640 I turn IS completely off. Normally in such a case I'm shooting a sporting event that can last for hundreds or even thousands of shots over several hours and there's no need to waste battery power running the IS when it is not needed. If I'm panning while shooting bursts the IS can sometimes fight the smooth movement of the lens. If the shutter speeds are lower than that, I usually turn IS on in Mode 2 (which is panning mode for my longer lenses with which I shoot using a monopod).

When shooting from a tripod, I do tend to turn IS off with shorter focal lengths that are less than, say, 70-100mm or so on a FF camera. However, if I'm using a longer focal length and there is any possibility of vibration while the image is being exposed, I turn it on. It can make a demonstrable difference, especially with very long exposures using a lightweight tripod and longer focal lengths. As always, experiment with the same lens you plan to use later, as this can vary from one lens model to the next.

If you do shoot with IS turned on using a tripod, be sure to activate metering and/or autofocus a couple of seconds before the shot to give the IS unit time to spin up the gyros and "settle in". I typically use mirror lockup with a tripod and longer lenses, so the 2-5 seconds I wait between locking up the mirror and firing the shutter via a wired cable release allows IS to "Stabilize".

What Canon doesn't tell you in the manual (that they probably should).

Another thing to keep in mind is that IS should always be turned off, so that the IS unit is actively "parked" in the "centered" position by instructions from the actively powered up camera body, before being removed from the camera and stored in a bag for transport. If the camera is in "sleep" mode, wake it up before turning IS off. Roger Cicala, the founder and Chief Lens Guru at lensrentals.com, once mentioned this, almost as an afterthought, in the description of a photo of the IS unit included in a blog article in which he and Aaron tore down an EF 70-200mm f/2.8 IS II and EF 70-200mm f/2.8 IS III to compare their construction and investigate what, if anything, had been changed mechanically between each version of the lenses.

Looking from underneath, though, you can see the plastic posts that we sometimes mention. These occasionally break, possibly from shock during shipping. If the IS is not turned off, the lens group is free to bounce in all directions with only the posts to stop the motion.

After being questioned for more details in the comments below the article he said:

The proper thing to do is 1) Turn IS off at the lens while the lens is still mounted to a camera. This 'locks' the IS unit in place. If you have IS on and just remove the lens from the camera, then it does not lock and off the camera flipping the switch does no good.

You can confirm by gently shaking the lens; there's very little noise if the IS is locked.

The locked position is safer for transporting the lens. If it's not locked the IS unit can bounce around and cause damage. How big a deal is it? I can't say for sure, but maybe 1 in 1,000 shipments that come back with IS not locked are damaged. But the incidence is 0 in 1,000; or very close to that, with IS locked.

While a tripod undoubtedly improves image stability, it doesn't guarantee it in all situations. There are other factors at play that become visible when the exposure is long. Take wind, for example. Even if your camera is on a sturdy tripod, wind gusts can cause noticeable vibrations ruining your image if the exposure is long enough.

In the city, moving transport like trains or trams can shake the ground and spoil your photo. Even you, the photographer, can do that while waiting for the image to be taken (especially if your camera is on a flimsy tripod extended all the way up). Again, I'm speaking about long exposures, especially when the ND filter is used.

So I'd suggest to switch the IS off if the weather is calm (to save your battery) or take a couple of images without/with IS and then choose the best one if the weather is windy.