Here in Northern California, roses don't really go dormant, but they need to be pruned once a year to get the best blooms. What happens if you don't? Well ... not much.
They won't die, and you won't be punished or publicly shamed by the rose police. But your roses won't be as good as they could be. And if you skip pruning for many years, the plants will become weaker because they never get that boost of new growth that pruning stimulates. In the wild, it's likely that roses get "pruned" by deer and other animals crashing through the underbrush and breaking off canes, or eating them.
January is considered by most locals to be the best time to prune roses in Northern California. Below I'm going to give you Julia's tips on pruning, mostly distilled from other sources, but if you'd like to read more or see a video, I recommend Rayford Reddell's classic book, The Rose Bible, or Rebecca Sweet's rose pruning video.
Start by stripping off all the old leaves. This makes it much easier to see where you need to cut, and also helps the rose go dormant for a while.
Cut out all dead canes and stems.
Cut off suckers (the canes growing from below the bud union at the base of the plant).
Cut off any canes that are smaller in diameter than a pencil.
Cut out crossed and highly crooked canes.
Cut back to whatever height you're aiming for overall (more about that below).
Cut each cane right above a growth node. That's the traditional advice, and I still follow it, but in truth there have been field studies showing this doesn't matter much.
The proper height to aim for is a subjective decision. You can find different guidelines for different classes of roses, and the more types you have in your yard the more you'll want to read. Here again I recommend The Rose Bible. But in general, you should take a cue from the way the rose is growing in the summer. Was it six feet tall in June, and did it shoot right back up to that height every time you cut off a long-stemmed bloom? Then don't prune it back to two feet. Take it down to about three or four feet and watch what it does. Observe all your roses all year long - get to know them. They have different growth habits. Some put up long straight canes and some branch endlessly into a tangled mess, always staying low to the ground. In my garden I have ‘Pink Promise,’ a rose that grows to eight feet tall every year, and 'Elena' is not far behind in height. But 'Amber Queen,' a lovely rose, prefers to stay under three feet high with lots of small criss-crossed canes. If I cut 'Elena' back too much, the blooms aren't as plentiful, and if I cut too many of the smallish canes out of 'Amber Queen' she spends her first few months trying to grow them all back, so I modify my pruning a little for each rose.