Last year we were embroiled in a project to purge Windows XP from our companies, a project that a more than a few people reading this are still dealing with. One of the methods that I've suggested over the years was to move the apps that only worked on Windows XP to Windows Server 2003, which is essentially XP Server, in order to buy some more time to figure out how to deal with those applications. If you took that advice, I hope you used your time well–Windows Server 2003 R2 is going End Of Life on July 14 of this year.
Even if you didn't use Server 2003 as a way to ease the pain of migrating from Windows XP, you may still have Server 2003 in your organization. I'm blown away when I run into a customer that's still using Presentation Server (usually 4.5, but does it really matter which version of "Presentation Server" anyone is running?). The story is usually the same: they let maintenance expire, and decided that since it did exactly what they needed it to do they'd just let it go rather than paying to re-subscribe to maintenance or upgrade. Those folks are in for a rude awakening!
The other big use case I've heard for running Server 2003 is to support 16-bit applications. Some people are shaking their heads saying "those don't exist anymore," while others are saying "really, others are having that problem too?" The challenge with 16-bit applications, as we all know, is that they won't run on an x64 server, so we're stuck running them on a much older 32-bit OS. There are currently two supported 32-bit server OSes from Microsoft: Windows Server 2003 R2 and Windows Server 2008 (R1, or whatever you call it). If Server 2003 is basically XP Server, that makes Server 2008...
...wait, what OS came after XP? I never really used it except for that one time to run the GeekOut at BriForum Amsterdam when a driver made it blue screen during the game...
...oh, now I remember...Vista! Yep, Vista Server.
That's a bleak outlook right there. If you have a 16-bit application that you simply have to support, your only options after July 15 are to either pay Microsoft for ongoing support or move your application to Server 2008. It's better than nothing, but it's not ideal.
It gets more challenging, though.
If you're a Citrix shop, you've got some issues to contend with. For instance, the the ONLY version of XenApp that supports Windows Server 2008 R1 is XenApp 5! XenApp 5 reached it's End of Life phase back in January, though Citrix offers extended support for it until January of 2020 when support for all versions of Server 2008 ends. There is no version of Presentation Server that supports Windows Server 2008 R1. All this boils down to the fact that if you have a need for a 32-bit OS, your only remote desktop option is Vista Server running XenApp 5.
So the options are limited when trying to use an RDSH platform for 16-bit applications, but there are things you can do to continue supporting those applications. As help4ctx kindly pointed out in the comments, Windows 7 (in fact, even Windows 8 and Windows 10) still come in 32-bit flavors, so they'll be able to run those ancient apps. If you want to continue accessing those applications remotely, you can do that with VDI.
So let this serve as a final warning. You have four months to remove Windows Server 2003 from your environment. You can go to Server 2008 R1, but all that does is set you up for more headaches. Don't upgrade to something that's already been placed on life support. Modernize your environment as much as possible, and this time keep it up-to-date!