If you had been able to turn it off right away, and put it in a bag of rice, I have seen that work for some people, at least for a limited time. There are claims out there of various methods.
But if the screen is bad, at minimum you have to replace that, that's virtually certain. You don't say what model it is, so not sure whether that's possible or how much it might cost.
Try the rice thing for a few days, you have nothing to lose at this point. When I used to take calls for Applecare I had several customers tell me this worked, the idea being the rice would suck the water out. I don't believe it's really a great idea but it can't hurt at this point.
Having said that, your phone is pretty much toast, per the other answer. When I was lent out to the Apple store, we saw Macbooks and phones come in where the customer said they had "dried it out" but "it wasn't working" a few weeks later.
Invariably we'd open the thing up and see some residual water in there.
Sorry to say, you are probably on the road to a new phone. :( Now if you try a third-party place to repair it, at least the last time I worked for Apple about a year ago, if you put third-party parts in there Apple will never touch it to do a repair. Unlike Macs where you don't "void the warranty," any third party parts in an iOS device precludes Apple from doing any repairs in the future.
So balance out free (rice) vs. third-party screen (cheaper) vs. shiny new iPhone 6S!!
To clarify a statement above, with Mac products, if you add a third-party hard drive, for example, or memory, PROPERLY, without damaging anything in the process, you can still take your Mac to an Apple store for service and warranty.
Again this was a year ago, the last time I dealt with it, but if an Apple Specialist or Genius opens up an iOS product and sees a third-party screen or battery, etc, they immediately stop working on it, in most all cases. You'd be amazed how fine an eye these Geniuses have for sniffing out third-party screens -- so I'd only go third-party (non-Apple) parts on something that's out of warranty.