You guys need to read Decision 13-2/2.
"13-2/2 Player Who Misses Tee Shot Presses Down Irregularities Before Next Stroke
Q. In playing a tee shot A misses the ball. Before playing his next stroke, A presses down turf behind the ball. Is this permissible, since the ball is in play?
A. Yes. Rule 13-2 permits eliminating irregularities of surface on the teeing ground, whether or not the ball is in play."
Then to support my point against allowing the player to press down the turf on the teeing ground of the next hole, see the definition of teeing ground:
Teeing Ground
The "teeing ground" is the starting place for the hole to be played. It is a rectangular area two club-lengths in depth, the front and the sides of which are defined by the outside limits of two tee-markers. A ball is outside the teeing ground when all of it lies outside the teeing ground."