no, most of the time, you don't have to reinstall linux.
however, there are still some cases you should be careful.
1)the cpu type is different, you cannot use a x86 linux on armv7
2)when the new board is too new, some features may not be functional.
3)if your linux is booted by /dev/sdXX, you need to make sure the hard drive is still /dev/sdXX
4)your video card may be different, but xorg will take care of it, just don't be panic.
The reason why you don't need to reinstall Linux because of the Linux Kernel of most distribution have included lots of drivers or auto detect during the startup. So even you replace the system with new hardware, generally speaking the Linux OS would still be able to boot.
But once you go further to start compiling your own Kernel, you'll have a more streamlined kernel based on your specific hardware, then you could still boot up your new PC with LiveCD and then compile a new Kernel again.