Tiny Core defaults to vga=791 or 1024x768 your machine may not be able to support that resolution.
At the first boot prompt type
tinycore xsetup
This boot code creates a selectable table from the Xvesa -listmodes so it is easier to use.
If you get a blinking cursor after the Xauthority lines, try hitting Ctl-Alt_Backspace to get back to text prompt then type xsetup.sh and try another screen resolution option. From text prompt type startx.
Note: Xvesa -listmodes displays what your video card is capable of, not necessarily what your monitor is.
If you system does not support Xvesa then perhaps framebuffer as was mentioned.
Last resort is to use the Xorg extension.