Does leaving your laptop powered on unsuspended and moving it around risk damaging the hardware?
Yes, there is a risk in leaving the computer on. But that doesn't mean you need to turn it off – it is enough to suspend (often by just closing the lid).
However, out of convenience I'd like to not turn it off when travelling between places.
Suspending is not the same thing as turning it off!
In suspend mode, most of the hardware is unpowered (CPU not generating heat, HDD heads safely parked), but the RAM contents still remain as they were – the OS is "frozen in time".
This means that the suspended laptop is safe to carry, but you can instantly resume working (even old laptops resume in 2–5 secs) – there is no inconvenient waiting for the OS to boot up again.
(You could say that the suspend function is specifically made for carrying laptops around.)
Note: If you leave the laptop suspended for a long time (several hours), Windows will eventually decide to hibernate to disk instead – to avoid running out of battery. (A suspended laptop still needs minimal power; depending on battery, it can remain suspended for several days or even weeks, but not forever.) Resuming from hibernate still retains all your programs, but is noticeably slower. So for convenience, you'll likely want to increase the hibernation timer to a few days.
I think it was the hard disk, or the memory he mentioned.
Possibly both. There are two primary sources of damage:
Electronic components (especially CPU, GPU, SSD flash) can be damaged by overheating. That's why computers have fans in the first place – but in a small bag/backpack there really isn't much airflow to cool anything down at all.
So if the computer in your bag is still running, in the best case it'll waste battery on fans. Worst case, the main CPU will soon reach its "emergency shutoff" temperature of 100–120°C and the whole computer will turn off anyway (to prevent further damage).
Meanwhile, devices with moving parts – especially the hard disk (if it's a magnetic HDD) – may be damaged physically due to sudden shocks causing the heads to crash into the platter, which in normal operation is a few nm away.
(Some laptops have motion sensors which try to detect when the device is falling and about to smack into the ground, but that doesn't provide anywhere close to total safety. They're not going to help much if the device is being constantly thrown around in the trunk...)
The suspend mode avoids all of these problems, because it does power off the CPU and HDD.
There are several different modes of operation for your laptop:
ON: the CPU is running, RAM is powered, the hard disk is running, the screen is on.
As the CPU is running, it generates heat (from a little to a significant amount depending on what you actually do), and this needs to be evacuated, usually with fans and vents. You don't want to have your laptop doing video encoding or 3D rendering while enclosed in a bag: it'll quickly overheat. But even when the CPU is "idling" it will generate heat and won't like very much being in a small enclosed space.
The hard disk (but see below) has a magnetic disk spinning at 5400 RPM with a tiny read/write head hovering just above the surface. You don't want that to take a shock while it's doing so, as it could damage the disk.
This is the state that draws most power, and is the most susceptible to issues.
Note that even in this state:
- the screen may be turned off while the CPU keeps running, usually after a period of inactivity. It'll turn back on as soon as you use the keyboard, trackpad or mouse.
- the hard disk may "spin down" (park heads, stop rotation) when unused for a period of time, and spin back up when needed. On some laptops you can definitely hear it stop and start.
- most hard disks are quite resistant to shocks, include fall detection (parking heads as soon as they detect free fall). Better not to tempt the devil, but we're not talking about eggshells.
- if you have an SSD (flash drive) rather than an HDD (magnetic disk drive with a spinning platter), then there's nothing to spin up or down, no moving mechanical parts, so no problem at all.
Sleep: The CPU and hard disk are not running, the screen is off, but RAM is still powered and keeping its contents.
In this state, there is just a little bit of power used to refresh RAM so it keeps its contents. That generates very little heat. There are no moving parts.
It is safe to put your laptop in your bag and carry it around. As there is still a bit of power used, the laptop can stay in this state for a while, but not forever. Some laptops will spontaneously switch from sleep to hibernation after a while to avoid running down the battery (either after a set time, or based on the battery level).
Hibernation: The contents of the RAM have been saved to disk, and then everything is turned off.
The only difference with the "OFF" state is that the state has been saved so it can resume relatively quickly afterwards (yawn).
Nearly no power at all is used (just enough to detect a press on a button and/or lid opening). It can stay in this state for as long as you like. It is safe to put in a bag or carry around.
OFF: everything is off. Same consequences as hibernation.
Depending on who you ask, "suspend" may mean sleep or hibernate, so you need to be more explicit to make the difference. But it doesn't change the fact that whether sleeping, hibernating, or off, a laptop is safe to put in a bag or carry around. That's the whole point of a laptop!
Final note: there are sometimes cases of laptops not actually going to sleep when you ask them to, or waking up during sleep and not going back to sleep. You can then find your laptop being quite warm when you take it out of your bag. Not good. But this is a specific issue, not a general rule.
To be completely fair, moving your laptop around when it's off does damage, too.
Manufacturers know this and it's why they try for stiffer frames, rubberized everything, tougher LCD screens, strain relief on internal cables, and so much more.
When I was a computer tech, we would tell customers that a desktop generally has an expected lifetime of 4-5 years and a laptop 3-4 years. I've gotten well over that with both, but generally it's true.
A laptop will see a bunch of abuse, no matter if you have the most plush carry case in the world. The bumps, jolts, moving the monitor, setting it on a table, picking it up, everything you do to move it damages it in some slight (or not so slight) way. Plugging in cables is much more often in a laptop than a desktop and can cause loose sockets.
Being on top of a desk, in a bag, in a vehicle, or wherever is much more likely to see an accident, too.
This is why I would caution people against buying a laptop. It's more expensive and more prone to damage than a desktop, so do you really need a laptop? If you need the mobility, then yes, you need a laptop. Just be sure you understand and accept the risks first.
Moving it while it's on is just more potential damage than when it's off. Even solid state drives aren't 100% guaranteed against all damage.
Edit:
I've been asked how simply picking up and setting down a laptop can damage a laptop. Well, there's the strain on any plastic pieces as well as the hinges. Depending on how people pick up or set down the laptop, there's torque put on the case and/or mobo. I've seen instances where a laptop quit working simply because it was picked up. I've destroyed my laptop by putting it in my cars passenger seat too hard, and I didn't even drop it very far. (It was the first time I had done anything like that, but it was old enough that any jarring was probably going to break something.)
The reason behind this is because integrated circuits (aka: ICs, or chips) soldered on the board don't stretch and bend at the same rate the fiberglass circuit board does, so eventually you can separate the solder points by these stresses. Heat cycles can do the same thing (ICs and fiberglass expand and contract differently, too), and in fact the heat of running the laptop can weaken the solder joints to make it more likely to happen if you are moving it while it's running. Surface mount electronics are just barely attached to each other at the best of times.
The plastic in the case can eventually break, leading to the stress being picked up going to the mobo or other electronics, instead of the case. The hinges can fail, causing the LCD to fall and crack, or simply wear the wiring/cables going between the screen and mobo, eventually causing a short or a break.
You can find lots of examples of all this usually irreparable* damage by doing a basic Google search. Parts can be replaced, if you can find them, are willing to pay for them, and willing to replace them yourself or pay someone to do it for you. I've seen it happen quite a bit over the several years I was a computer tech, and several times since I became a programmer.
(*) People try to repair circuit board damage by reflowing the solder with an oven, hair dryer, or heat gun to varying success. Usually the repair is temporary, even if it's successful at first.