The short answer? There isn't really a way to open an .irx file without the device originally intended to open it. There are two main uses for .irx files-- GPS devices and PlayStation 2 games.
Both instances rarely involve the use of a home computer, as the .irx file is associated with a proprietary device: either the GPS device or the PlayStation 2. DeLorme GPS devices using the application Xmap also rely upon .irx files.
The only computer application for an .irx file would be those making "home brew" games to run on their PlayStation 2. There is an older application for IBM computers, where .irx files were used to create rules for the operating system. However, this application is obsolete, so it is unlikely you are looking to open an .irx file for that application.
So why do PlayStation 2 and GPS devices use .irx files? An .irx file is a file that has both executable code and data within. This means the file can make the computer or in this case, game console or GPS device, do something with something. As most video games are user dictated actions in an environment designed by the developer, you can see how this would be a good "nuts and bolts" component. GPS devices now have quite a bit of interaction with the user as well, so an .irx file is great fit.
An .irx file, which stands for Interactive Resources Executive, is really a modified ELF object file. No, this isn't an elf like a being with cute ears. ELF is another acronym, standing for Executable and Linking Format file. ELF files work on Unix machines, like many popular game consoles today. Windows based computers are not Unix systems, and therefore cannot read Unix files.
ELF files have a few basic components. First, there are segments that are implemented and used during execution, or run-time. Second, there is data stored in segments used by the first segments to change conditions during the execution, or run-time. Instead of data, the second section may also include links to locations of data or other files. This function makes up the "linking" aspect of the file name.
There is not a readily available way to convert .irx files into another format readable by another application. In one way this is good, because 99 times out of 100, an .irx file is associated with proprietary software. This means in all likelihood, you shouldn't have access to the .irx file. In the case of "home brew" PlayStation 2 games, the .irx file would be opened by the game you develop, or worked on with the compiler you are using.