Azure disks are stored as page blobs in Azure Storage. A storage account has specific scale targets. As you drive heavy disk I/O load through a VM, it's possible to push the limits of these storage targets. Locally-redundant storage accounts have a perf target of 10Gbps ingress/15Gbps egress, while geo-redundant accounts have a target of 5Gbps ingress/10Gbps egress. A specific blob (which holds a single disk) has a target of 60MBytes/sec.
This is why you will see suggestions (such as that TechNet video) for using multiple disks across multiple storage accounts, to go beyond account-specific storage scale targets.
Here's an article regarding VM sizes and related max IOPs. You'll see that each VM size has max. IOPs equivalent to 500 x #disks attached.
This article covers performance considerations for running SQL Server in a virtual machine, that covers many topics including disk IO.
EDIT This answer is a bit outdated now. Aside from standard storage disks, you may now use Premium storage for disks. Premium storage is SSD-backed, and disks range from 500-5000 IOPS, and 100-200MB/sec, depending on disk tier (the highest being 1TB, 5000 IOPS, 200MB/s). You will still have overall limits for a given DS- or GS- series VM, but these limits are higher than D- or G- series (maxing out at 80,000 IOPS and 2,000 MB/s on a GS5, for example).
More info on VM sizes here.
More info on Premium storage here.