Executing same functions for different classes

Couple things:

First off, you should always wrap if statements in { } curly brackets.

Object part = json.get(0);
File saveTo = new File(dataPath("test.txt"));
if (part.getClass() == JSONObject.class){
  ((JSONObject)part).save(saveTo, "");
}
if (part.getClass() == JSONArray.class){
  ((JSONArray)part).save(saveTo, "");
}

Secondly, if you only expect one of those if statements to execute, then you should probably use an else if:

Object part = json.get(0);
File saveTo = new File(dataPath("test.txt"));
if (part.getClass() == JSONObject.class){
  ((JSONObject)part).save(saveTo, "");
}
else if (part.getClass() == JSONArray.class){
  ((JSONArray)part).save(saveTo, "");
}

Third, you could use the instanceof operator instead of getting the class:

Object part = json.get(0);
File saveTo = new File(dataPath("test.txt"));
if (part instanceof JSONObject){
  ((JSONObject)part).save(saveTo, "");
}
else if (part instanceof JSONArray){
  ((JSONArray)part).save(saveTo, "");
}

But to answer your question, there isn't a great way to minimize this code, because JSONObject and JSONArray are both direct subclasses of Object. It would be easier if, say, JSONArray was a subclass of JSONObject, but it's not.

But if you find yourself writing this code in multiple places, you should extract it into a function, like this:

void saveJsonThing(Object part, File file)
    if (part instanceof JSONObject){
      ((JSONObject)part).save(saveTo, "");
    }
    else if (part instanceof JSONArray){
      ((JSONArray)part).save(saveTo, "");
    }
    else{
       //handle error?
    }
}

Then your code can just call that function whenever it needs to save something:

Object part = json.get(0);
File saveTo = new File(dataPath("test.txt"));
saveJsonThing(part, saveTo);