If I use all my Strike teams on a tactical strike map, is that it, can I no longer play those maps until I start a new campaign? Will I no longer be able to play any tactical strike maps until I get new strike teams? For that matter, when do I get more strike teams?
Three strikes and you're out. If you get what I mean...
Anyways, you get more strike force missions as you go further into the story. The campaign does continue on but the success/failure of the strike force will affect the rest of the story. As in, you get new strike teams for later strike force missions.
The Strike Force missions include:
[missions from Call of Duty wikia]