If there's a Lich in a cave nearby and your character kills it, then the guy 5 minutes behind you can't kill it again for his quest because it's already dead!
Do not do this. This opens the door to player griefing (moreso if there's a sellable drop on the same enemy).
Personal experience - In Final Fantasy XI I wanted to get the quest item for the Paladin Job, which drops off of a certain ghost that spawns about 5 or 6 times every night. There was someone standing there all day waiting for it to spawn and killing it as soon as it popped up (I suspected botting, but I couldn't prove it). The only way I could get the item was to buy it off the Auction House.
Even if a tradeable item is not involved, there is a subset of players that will abuse the system just to piss others off.
Guild Wars 2 achieves a nice balance with this - there are events and bosses that sometimes appear on the map that you can take out (most of them require a group, but not all, and they scale) but at no point are you required to complete it for a quest. Success/fail of these missions chain into other events triggering - causing a bit of a domino effect.
Now granted - the idea of smaller servers that die and start up kind of mitigates this, and there is a market for town/empire building games using this structure;
I remember when Batheo first started, my guild was on the top tier. They start a new server every other week, I think (they are up to 68 servers!) The problem with it is that servers 'die' as players leave - no one wants to join an 'old' server and players leave to join new servers. The business model is based on selling 'perks' in order to build faster or pay to have an advantage - so each new server there are more ways to pay to have an advantage and get ahead before other players do. You can play for free, but you'll lag behind a little unless you spend all day at it (and even then, there are those who pay for the edge AND play all day)
This business model is popular for games developed in China and Korea.
Here is another one that's quite popular;
In this one, you work as part of one of the Three Empires and you can purchase buffs that benefit all players on your team and send armies to capture cities. They are up to 98 servers now (I think they start a new server every 3 weeks or so?)
I haven't seen this model used for an adventure MMO like WOW, but it might be worth checking out games that use something similar to what you are describing and see what is worth improving on.