How to quickly fix botched latex paint area on exterior door?
I wouldn't try to scrape off the messed up area while it's wet because you'll leave ridges at the edges of the area, which will look even worse.  Leave the area to dry before trying to fix it.

To get it to dry quickly, higher temperatures and low humidity are your friends.  I've used a space heater and an oscillating fan when I needed paint dried quickly: arrange the heater and the fan so that warm air gets blown past the work piece continually.  Leave it running overnight: it looks like you're using a higher-gloss paint, which will form a skin quickly, but underneath will remain wet for a while.  

(See here for a technique for removing drip marks.)