A quick google shows these devices widely touted as "free energy" or "overunity" devices - as such you are correct to view them with a lot of suspicion. Clearly any energy is not going to be free - we can expect the output to be lower than the input ... so you question seems to be more along the lines of:
Can we make this setup turn mechanical work into electricity?
More broadly: what would we have to do with this setup to make it turn work into electricity?
You could build one to see... sounds like someone has already discussed this here:
https://www.physicsforums.com/threads/toroidal-inductance-generator.331215/
... didn't end well.
You can imagine a section of a toroid coil with a magnet inside on wheels, rocked back-and-forth like a rocking horse ... that would do it by the same principle as the "Faraday torch". The wheels mean that the magnet tries to stay at the lowest point while the torus-bit moves about it.
Care would need to be taken about the coupling between the magnet and the coil or they will just end up in sync. (The current induced by the magnet sets up a magnetic field inside the coil which opposes the motion... tending to drag the magnet with the coil.)
But that's just a quick naive impression off the top of my head... I think we'd need a detailed proposal to examine, and I have probably missed something.
To get a feel for the way crackpots misuse electricity and magnetism, and charlatans exploit common misunderstandings, you should look at D Simanek's impossible devices page.
https://www.lhup.edu/~dsimanek/museum/unwork.htm
... he has examples of magnet and electric "machines", and puzzles, that illustrate problems.
So much for my own resources...
If the toroid design were good for generating power, we'd expect to see some in industrial use ... so it is a good idea to see what others have tried.
Unfortunately the field is dominated by the overunity/free-energy crowd.
I found two patents:
http://www.google.com/patents/US20130162057
... this has two solenoids with the magnet in a torus tube that passes through both.
http://www.google.com/patents/US20120235528
... probably closer to what you are thinking about.
The existence of a US patent does not mean the devices work as advertised - both these designs feature on overunity websites so it's safe to dismiss them right away.
More academically:
http://web.mit.edu/kirtley/kirtley/binlustuff/literature/electric machine/TORUS.pdf
... description of a generator/motor that uses a toroidal stator: with mathematics.
Seems these guys have published in IEEE Vol139 Iss6
http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=170913&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D170913
Intregued, also found a paper (same journal) studying toroidal generators...
http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=641924&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D641924
Comparing these designs with those in the patents: they are much more complicated, with much tougher maths.
If the patented designs were any use, wouldn't these engineers studying the design principle have come up with something similar and saved themselves a lot of work?
... so the short answer seems to be "yes and no" - a torus coil can be used as a component in an electric generator but the kinds of configurations used in youtube free-energy vids are not useful as normal generators either.