We have several Visual Studio subscriptions, which we use for development purposes. Are we allowed to run this software on Azure Virtual Machines?

Go to manufacturer overview

Yes. You are allowed to run any subscribed software, except Windows and Windows Server, on Microsoft Azure Virtual Machines.

Sources:

Microsoft MSDN subscriptions Software usage rights (https://msdn.microsoft.com/en-us/subscriptions/cc150618.aspx), as of 09.03.2017

Whitepaper_Microsoft Visual Studio 2015 Licensing, Stand November 2015

Whitepaper_Microsoft_Visual-Studio-2015-Licensing-1Download