Does a cow need to keep being pregnant to keep giving milk?

No. Cows can (and commonly do) lactate without being pregnant. A cow can be open (or not pregnant) for several years and keep giving milk...as long as she is being milked out regularly. If not, then she dries up and then you will need to breed her to get her producing milk again.

Dairy cows are bred so that they keep producing offspring or replacements for the dairy herd, mainly because the culling rate (or turn-over rate, if you want to think of it that way) is quite high--some farms have a cow turn-over rate of around 30 to 40%, depending on management criteria. As such, replacement heifers are needed pretty well all the time to replace those cows that get culled.