Depends how you define "teamwork"
Take TF2 for instance, are the medivacs and heavy casses really doing "teamwork"? they tend NOT to work together.
You see, the heavy class tends to just walk in, leading the way shooting up whatever depending on the medic, the medic will follow the heavy class, but there is absolutely NO coordination between the 2 still, no corner spotting for each.
In cs 1.6 there is teamwork within high class teams, whereas you will circle opponents, force them bunched up and try to flash up then run in from several sides.
THIS requires coordination, coomunication (in essence TRUE teamwork)
Now let's look at another example battlefield bad company 2.
Here they have added "spotter" function, "barrage" function etc, which makes you help your team automaticly, giving the illusion of "teamwork".
But let's face it, when you're doing this stuff you aren't actually coordinating and communicating with team8s, your just following the basic guidelines of the pointsystem of the game.
OFC you can play bf bc2 with true teamwork, but it's not what you will see in a pub.
To enforce true teamwork in a game there's 2 things needed:
1: high skill ceiling, which will encourage people to team up with people of their skill level
2: Reduction of what I'd call "solo effects", such as chopper gunner in cod mw2, whereas if you die while using it it STILL doesn't matter, it would be so much better if that guy using it was forced to posistion, and be defended, and could even put it down and help shoot when his team8s told him to. (easy way to do this would be a "cancel upon death" effect
3: Long-distance survivability, this actually really matters for teamwork, making it hard to successfully shoot som1 long-distance will help teamwork effort immensely.
Ofc its important to keep it quik on close combat quaters, therefor this adds to point #1, shooting MUST be difficult in order for teamwork to arise.
(long distance shooting, means you can pin someone down etc, at least lock a corridor/slow it down, while still staying alive)
4: Aoe effects MUST never be to strong! I think both BF and COD games does this mistake a lot, if you have to strong/easy to use aoe effects, any grouping up wil be a mistake.
I will say this only once, AOE effects encourage one-man-army-play.
5: offense/defense play, some sort of offense/defense play is important for teamplay, deathmatches have a tendency of encouraging what i like to call random-camping, where-as you camp in a comletely random insignificant spot, mainly because you not participating does NO damage, and if you can get that 1 kill you will have made up for your life that round anyway.
Capture points/bomb-play/rescue/reach point etc are all good solutions in my eyes, but personally my favourite would be in a 6v6 situation for instance to have a "at least 2 out of 6 must reach spot X" which opens for most strategies (sacrifical decoys etc)
Il end it with this: strategies is what encourages teamplay, open up for strategizing and teamplay will soon follow, one of the shooter games i experience most teamplay in in random pubs must be Natural selection 2.