All injuries that could hamper a player's play or their ability to play must be reported according to NFL guidelines. The NFL has also added an extra level to injury reports by measuring a player's participation in each practice.
Do coaches report everything? No way. Look at Andrew Luck this year. You don't think if teams knew he had busted ribs he wouldn't be blitzed half the game?
What do teams do? Well most of the time injuries will surface from other sources. There are beat writers that will mention someone was stretching their hamstring a lot and not practicing. Something major is very hard to hide. It is even a lower chance that there is a significant injury that can be hidden but the player will also play.
Joe's answer is right on that the need for an injury report stemmed from gambling and insider activity. Imagine massive wagering based on knowing your first two quarterbacks were out - as the team janitor. Team janitor could just sell this information too.
What I don't think is stressed enough is the impact of revealing injuries on a weekly basis. It makes keeping a healthy team even more important. It is a true double-whammy. Your player doesn't play as good because he is hurt PLUS the other team plays (him) differently because of the injury.
The easiest example I can think of is a WR dealing with ankle or foot issues. Whatever leg it is on you know they will have trouble pushing off with that foot - limiting their route tree. You also know that a quick slant will be harder to run. What do most teams do in this situation? Hard-man press on this WR. Perfect example? Sherman on Dez Bryant coming off broken foot. There is no way Sherman plays Dez that tight (Dez is a big strong boy) if Dez were totally healthy. The Cowboys even trying to throw to him at all was laughable and might have cost them the game. Not only was Sherman smothering him but it took him 2-3 seconds to work the first 10 yards.
What can teams do? Lie. Hope the injuries don't get out. Make the injury seem more insignificant than it is. Make the injury seem more significant than it is (Patriots).
Would I lie as a coach in the NFL? Hell yes. I get paid to win not to help the league in their gambling endeavors. I would seriously consider that 20-50 games a year are directly impacted by injury reports. The NFL is full of close games. Knowing the opponent's weaknesses might be enough to sway a game 5-7 points.