It depends on your interpretation, regarding tinnitus. It's shown a few times by playing a high pitch sound. There's a few scenes where his brother tells him he needs to protect his ears or it will get worse (and one with a Dr telling him), and there's a part he goes to a hearing convention and doesn't admit to his wife where he's going. I suppose this shows the embarrassment side. I liked it, but it is quite a slow paced movie. However, it's very emotional.
There's another scene where he tells a Dr/counsellor how he got tinnitus, and how he tried to hang himself (he doesn't verbatim link the tinnitus to the hanging but it could be inferred). The reason my opening sentence says it depends on your interpretation is because you could easily link the tinnitus to his prescription drug and alcohol abuse problems, or you could see them as separate issues. Having tinnitus myself, my interpretation is that his ear problems were driving him to get drunk 24/7, but I'd say the average viewer would not make this link from the way the movie is presented.
He doesn't show signs of being afraid of loud environments, but then, that's not always the case in real life, and since he's a rockstar, it's believable.
Overall, his ear issues are definitely a part of the plot. Just don't expect to see him constantly complaining about his ears though. It's inferred they are getting worse when he keeps saying "what" and has to lean in during conversations. Very similar to how Paul Gilbert behaves in his interview about tinnitus.