How can I pull up tack strips nailed to concrete?
Jim and Taylor are correct.

Use a small pry par because it is not as thick and will get under the tack strip better. Hold the the long straight part or the pry bar and place the curved end so that the but is against the floor and the wedge part is up against the tack strip directly in front of the nail. Whack the the curved part of the pry bar with a hammer, it usually only takes one or two hits to pop up the nail. 

It helps to hold the bar at an angle so the bar goes under the strip and to  not hit to hard, sometimes you can drive the end of the bar into the wall or trim if you get to aggressive. 

Do not pry up once the nail is free as this will splinter or break the strip at the next nail location, just continue whacking all the nails and when you get them all the whole tack strip will be free and easy to dispose of.