The flatter, wide bladed grasses with deep roots like Coarse and Tall Fescues and Rye and St Augustine are generally more wear (and drought and urine) tolerant, probably followed by vigorous grasses like Bluegrass and Zoysia, with short, fine bladed grasses like Fine Fescue and Bermuda probably least suitable. Also, longer grass will handle wear better than very short-cut grass, partly because it has more blade area to collect solar energy, and also as longer grass reduces nail damge to the root area. Try to rotate their play area around, too, so one area does not get worn out too much and it has a recovery time.
Maintaining a regular, moderately frequent watering cycle will help reduce yellowing from the urine, by washing the excess nitrogen down deeper into the ground - geting your soil checked and applying ground limestone (the gray garden lime - NOT hydrated lime, which will burn the grass and kill it) if called for also helps neutralize the urine.
Find lots more hints at articles like this one -
and by goolging "best grass for dogs" to read up on lawn and garden website articles.
If you have one place they wear out consistently (typically between the dog door and their bathroom), you can keep them from cutting across the grass with a bit of training, and using 2x2 (cedar best for rot resistance) wood posts with a hole drilled near the top, or metal pin type posts with an eye in the top, driven into the ground a foot or so (make sure not to hit gas line or electrical/communications/TV cables) about every 6 feet, then run a rope, vinyl-coated clothesline cable, or decorative chain through them to make a single-level barrier about a foot off the ground (less for small dogs) - then train them not to cross the barrier. Obviously, easier to train them in a new house then when they have already been cutting across the lawn for some time, but most dogs catch on pretty quick if you walk them along it and say NO every time they try to duck under it or jump over it. I use 2x2's with a 6" piece of 1-1/2" copper pipe hammered down over the top (so post does not split when driven into the ground) and drilled with a 1" hole to accept a small chain - that way I can remove the chain in the winter so the dog can play in the back yard in the snow without restriction. I remove the chain once there is about 4 inches of snow on the ground that looks like it is going to stay, so the grass is protected. This method also works to keep a fairly well trained dog out of flower beds and planting areas, and using a chain or cable with snap hooks on the end makes it easy to let down and pull out of the way for mowing.