If I appeal, can the appeal be dismissed?

The appeal can be dismissed if you do not follow the rather strict rules that govern appeals or if the appellate court concludes that the appeal is essentially frivolous.  

You can also have your appeal dismissed if the case becomes "moot". If the court can no longer give the requested relief, the higher court will dismiss the appeal. For example, assume you have filed the appeal of an injunction prohibiting a strike. If the strike is settled before the appeal is heard, the higher court would dismiss the appeal as being "moot". 

Just as there can be frivolous lawsuits filed, there can also be frivolous appeals. Most legitimate appeals rest on a clear legal error that is built on well-established law. An example of a legitimate appeal would be an appeal that is due to a judge giving an inaccurate instruction to the jury that effected the jury’s final decision. Conversely, litigants can also request an appeal where it is unnecessary. Typically frivolous appeals will be made for inconsequential errors that judges make, e.g. - for a specific objection decision that the appealing party disagreed with, attorneys often object to the admission of certain evidence and then, if it is admitted, appeal the judges decision later. If the court decides that an appeal was frivolous and dismisses it, they will also typically add interest to the judgment for the time of the appeal. 

On rare occasions, a case is settled after the trial but before the appeal. In these cases, the appeal becomes moot, or non-applicable. Typically, appeals dealing with injunctive relief tend to be dismissed as moot, as one party will act after the judgment is rendered to prevent the appeal from taking place. For example, if an environmental group was suing for an injunction against logging on some private land and the court rules in the logger’s favor, the loggers may simply hurry and complete the job before being served with the appeal papers. This would render the environmental group’s appeal moot.