Answer: $763$


Three positive integers that multiply to make $2009$ are:

$1\times1\times2009$ has faces $1\times1$ (two faces), $1\times2009$ (four faces). Not enough stickers for the $2009$ face.

$1\times 7\times 287$ has a face which is $7\times287=2009$ - not enough stickers.
$1\times 41\times 49$ has a face which is $41\times49=2009$ - not enough stickers.

$7\times7\times41$ has faces $7\times7$ (twice) and $7\times41$ (four times)
Surface area $2\times7\times7+4\times41\times 7 = 1246$
 
This leaves $2009-1246=763$ stickers left over.