Storage and network
Lots of comments but few people actually discussing the question being asked (a reminder: “How is desktop virtualisation likely to impact my existing network and storage infrastructure)
For larger enterprises, VDI is often the straw that breaks the camels back when it comes to storage infrastructure. Desktop workloads are vastly different to server workloads (e.g. logon storms) and many companies who have utilised existing storage that has quite happily being hosting their virtual servers find it now can no longer cope. This either requires expensive storage upgrades, or addition of some of the new IOPS "sink" technologies like Whiptail or Atlantis ILIO (still both expensive)
Another option is to utilise local storage and leave your expensive SAN/NAS for your virualised servers. Kaviza VDI-in-a-box (now Citrix VDI in a box) is a good candidate for this, as it uses commodity hardware, and just scales out using local storage as you need to add capacity.
Network-wise, the increased demand for storage bandwidth (if you aren't using local) may force you to investigate 10Gig ethernet. VDI is also 100% network dependant, so having reliable WAN and Internet links is paramount. There's no "offline working" scenario with hosted desktops, so multiple-resilient links are a must if you have business criticial offices connecting to centralised VDI infrastrucutre. And they don't come cheap.
Also, if you're delivered a "rich user experience" including videos on your VDI infrastrucutre over the WAN, then you might want to consider WAN acceleration and caching devices such as Riverbed or Citrix branch repeater.
How much you need to invest/upgrade will depend on the size of the organisation and the product sets you choose. It's a minefield, and can easily blow up in your face (hence the icon), so be careful out there.