After 3 years, the registration will be taken off the system is true, you'd have to re-register, and, get a new plate even.
but the fine is only 1000 baht if you don't surrender the plate upon 3 years plus back taxes (1% interest per year)
to re-register, various fees are around 1000 baht
this is all if you do it yourself, at the DLT, preferrably less hassle if you can go to the original DLT that the bike was registered in, if not, you have to transfer to the new DLT, and more forms and fee..
so, if you go to the agents that can do it for you, charging around 4000 baht for the convenience doesn't seem for off
you can pay the road tax and get the necessary insurance online now though, they'll mail you the new tax paper, so you'll need a Thai address