While slightly different from your situation, I had a computer which, when it got to the grub screen, if I pressed any key the computer would lock up.
This would happen with any operating system, so it was a computer problem, not an operating system problem.
As a result, I could only use one operating system on it.
I can't be sure of the cause, but the motherboard had connections for both IDE and SATA. I used SATA hard drives. IDE drives were the primary drives, SATA were additional.
I assumed that if I used an IDE drive it would work correctly, but never tried it. I just used one operating system.
The motherboard has since died, and I trashed the computer, so it is history.
Anyway, one thing you can know, if it happens with all operating systems it is a computer problem. If it only happens with Tiny Core, you can post that info (after you have checked the previous post).