Can I cause damage to electrical appliances by unplugging them when they are turned on?
Typically, an appliance will not be damaged if it is currently ON and then you unplug its power cord. If you plug it back in the device will simply resume operation as if it were ON.

I say typically because there are some cases where this may not be the case. There are many devices such as computers and some other electronic devices that almost always recommend that you interact with the device to gracefully shut it down before you think of yanking the power cord out of the wall. This graceful shutdown process ensures that the device has finished all critical internal operations before losing power. This could include operations such as closing open files or saving important system state to a storage memory. 

With many of today's appliances having more and more smart electronics, it can be more of a concern that you not arbitrarily pull power cords. A well-designed appliance from a reputable manufacturer will have made provisions in the electronics design and software so that the device does not get bricked when the power is suddenly lost. On the other hand, as more designs are done by less experienced people, this possibility of damaging operation of an appliance by the sudden loss of power is more likely than ever.

To make my case, let me comment that the ever more popular Raspberry Pi (rPi) microcontroller board is leading many people to think of embedding devices like that into appliances and other gadgets. It is a known fact that removing power from an rPi board can brick the boot media on its micro SD card and keeping it from being able to reboot when power is restored.