How can I remove a portion of a hardwood floor board?
For close cutting, either horizontally or vertically in very close quarters, I  recommend a multitool.



It has different blades for wood, metal, and can do some limited grinding and tight sanding. You do need clearance of at least the width of the blade plus about 1/4 inch. 

If you do not have that much clearance, you may need a Dremel-type rotary cutting/grinding tool. It can cut or grind down to very small widths.

 

Failing that, or if you are hand tool inclined, you can use a chisel.