do I need to surround fileInputStream.close with a try/catch/finally block? How is it done?

Yes, that is the common pre-Java 7 solution. However, with the introduction of Java 7, there are now try-with-resource statements which will automatically close any declared resources when the try block exits:

try (FileInputStream fileIn = ...) {
    // do something
} // fileIn is closed
catch (IOException e) {
    //handle exception
}

Because FileInputStream.close() throws an IOException, and the finally{} block doesn't catch exceptions. So you need to either catch it or declare it in order to compile. Eclipse's suggestion is fine; catch the IOException inside the finally{} block.

The standard approach is:

FileInputStream fileInputStream = null;
try {
    fileInputStream = new FileInputStream(...);
    // do something with the inputstream
} catch (IOException e) {
    // handle an exception
} finally { //  finally blocks are guaranteed to be executed
    // close() can throw an IOException too, so we got to wrap that too
    try {
        if (fileInputStream != null) {
            fileInputStream.close();
        }        
    } catch (IOException e) {
        // handle an exception, or often we just ignore it
    }
}