One of Game of Thrones most popular sayings is, "Valar morghulis," meaning, "All Men Must Die," but does that include dragons? Thus far, Daenerys' three dragons have only gotten bigger and stronger, and haven't really suffered through much pain of any kind. However, during Daenerys' onslaught against the Lannister army, Drogon was hit with a large spear. Though the injury wasn't fatal, it did concern a lot of fans. So will Daenerys' dragons die on Game of Thrones? It's definitely a possibility.
The arrow Drogon was hit with was part of a new weapon that was given to the Lannisters, specifically to attack Daenerys' dragons. While the hit wasn't nearly as harmful to Drogon as Bronn, who shot the arrow, hoped it would be, it was still successful in that it grounded Drogon (thereby making him more vulnerable), while most regular arrows had no impact on his tough scales. Had the arrow hit Drogon in a more vulnerable area like the throat or near his heart, it could've actually had dire consequences.
Moreover, if Cersei has more of those weapons made (Drogon quickly blew up the one that was used to hit him), they could easily kill at least one of Daenerys' dragons. Though her dragons are certainly strong and obviously their fire-breathing is their biggest asset, if they were being shot at from different sides, it would be difficult for the dragons to avoid them all. (They are pretty huge targets, after all.)
Additionally, there's a theory going around, based on the books, that Euron may be in possession of a dragon horn. The magical horn gives the user the ability to control dragons, and in the novels, Euron was in possession of one. If Euron has one on the show, it'd make sense that he'd hand it over to Cersei. With the horn, Cersei could easily turn the dragons on each other and on their mother, Daenerys.
However, if Euron really did have this horn, why hasn't he already mentioned it to Cersei? It seems more likely that Euron doesn't have it, but that doesn't mean it doesn't exist on the show. If Cersei does somehow come into possession of a dragon horn, that could definitely change the tide of this war again, in Cersei's favor.
For now though, Cersei's best chance at killing Daenerys' dragons is her large arrow contraption, and even with that her chances still aren't that good. At this point in time, Dany's dragons seem to be relatively safe, but anything can happen in the world of Westeros.