Monsters cannot have negative ATK or DEF values. If an effect would cause a monster to have a negative ATK or DEF, that value becomes 0 instead. Effects that modify ATK and/or DEF values always constantly apply themselves (except for some effects that specifically state otherwise).
Example: Koitsu has 200 ATK, and is a Level 10 monster. Your opponent activates Burden of the Mighty. Koitsu would lose 1000 ATK, making its ATK -800, but this is negative and can't happen. Instead, its ATK becomes 0. If Koitsu battles your opponent's Berserk Gorilla (2000 ATK), you would lose 2000 Life Points and NOT 2800, because Koitsu's ATK is 0 and NOT -800.
Later (assume the battle didn't happen, or that Koitsu somehow survived it), you equip Phalanx Pike to Koitsu (with 1 other Koitsu in the Graveyard), which provides +900 ATK. This does NOT make Koitsu's ATK 900, because now Burden of the Mighty can subtract some more ATK points (it had only subtracted 200). Koitsu's ATK is now 100 (= 200 original ATK +900 Phalanx -1000 Burden).
Read the ATK and DEF modifiers (the ones that subtract, of course) as saying "subtract as much as possible of xxxx ATK/DEF without letting that value become negative". Something like that.
Secondly, and a very important point to make, is that monsters are NOT destroyed when they should have negative ATK/DEF. There's nothing in the rules to suggest this. As far as I know, only Venom Swamp destroys monsters when their ATK reaches 0 (or less, since negative values automatically become 0).