Can the ipad3 print using the HL-2270DW from Brother?

The list of AirPrint printers can be found on the AirPrint 101 page.

It's not one of the models that support AirPrint (they are listed on this page), and Brother's iPrint app doesn't appear to support that model either. You could try one of the third-party printer apps such as Print Central For iPad or Print n Share

Since your printer isn't on the list of AirPrint capable printers, http://support.apple.com/kb/ht4356, your options are as follows:

1. Download the app for your printer manufacturer or a printer app that will print to several different makes of printers such as Print Magic, Printer Pro, etc. In order to print you must send documents, photos, etc. to the app.

2. Load a program on your PC/MAC that will allow your shared printers to appear as AirPrint printers. A few programs to look at are Printopia, AirPrint Activator, FingerPrint by Collobos.

How to Print from Your iPad: Summary of Printer and Printing Options

http://ipadacademy.com/2012/03/how-to-print-from-your-ipad-summary-of-printer-an d-printing-options

Print from iPad / iPhone without AirPrint

http://ipadhelp.com/ipad-help/print-from-ipad-iphone-without-airprint/

How to Enable AirPrint on a Mac and Use Any Printer

http://ipadhelp.com/ipad-help/how-to-use-airprint-with-any-printer/

 Cheers, Tom 😉