I have a Shadow Warrior HE unit which says something like "Forced - at the beginning of your turn, each opponent must assign a pt of dmg to a unit in his battlefield." Is there any way that opponent can respond with a tactic such as "sacrifice a unit in your battlefield that was probably going to die anyway in order to do a really cool effect."

To be more clear, I mean play this tactic in direct response to the forced effect in order for it to happen first on the chain. Or instead, in some action window in between assigning and applying dmg - even though I don't think there is any such window exists in cases such as these, but I'm not completely sure.

No. Forced effects resolve before any action can be taken. See Time structure in the FAQ

Thank you. You've been most helpful sir.