Copying contents of two files into one file/ bufferedReader close calls
A little bit about code style: it is conventional to surround binary operators with whitespaces(for instance, getApplicationContext().getFilesDir()+"/BirthdayReminders/fileone.txt" should be  getApplicationContext().getFilesDir() + "/BirthdayReminders/fileone.txt") and there should be a whitespace after a closing bracket and before the opening one(that is, } finally { is better than }finally{ and so on).
Yes, you should check that br and r are not null before calling the close method on them. The FileReader constructor can throw an exception. In this case they will be null.  However, you can use try-with-resources(if you have Java 7 or later) to make your code more simple and concise:

try (
    BufferedReader reader1 = new BufferedReader(new FileReader(...));
    BufferedReader reader2 = new BufferedReader(new FileReader(...))
) {
    // do something
} catch (IOException e) {
    // do something else
}

Variable names: r and br are not really descriptive(however, their scope is very narrow so it is not a very big deal in this case). The same holds true for other variables: I would use something like lines instead of list and so on.
This kind of initialization is pointless in my opinion:

BufferedWriter writer = null;
writer = new BufferedWriter(new FileWriter(getApplicationContext().getFilesDir() + "/BirthdayReminders/output.txt"));


This one looks better:

BufferedWriter writer = new BufferedWriter(new FileWriter(getApplicationContext().getFilesDir() + "/BirthdayReminders/output.txt"));

The listWord variable and an index-based loop is redundant:

for (String listWord: list) {
    // do something
}


For-each loop is shorter and simpler.