Is there any reason to actually hurry?

Solution 1:

Most of the time, no.

However, there are at least two scenarios where you'll end up failing the mission if you don't keep moving:

The nightclub shootout in episode 2. If you spend too much time poking around after falling/shooting through the VIP Room glass, you'll fail.

In Episode 6, after the bomb goes off. If you don't keep progressing through the inferno (at least during the first part where you go through the doors and up the stairs) you'll also fail there as well.

Solution 2:

No, at least not to my knowledge. Just like in every other game that encourages exploration (via collectibles, achievements and the like) it does not punish you for it. You can take your time and explore as long as you like.
The only special case in Max Payne 3 are the additional modes like New York Minute, which limit the time available to finish a particular level.

Many games tell you it is utterly important that you do something quickly but allow you to wander around aimlessly for as long as you like. This is quite comical indeed. Another recent example of this is Far Cry 3.