Update
You can now turn it off in the portal in the Web App settings. You can also use the resource explorer as described in this blog post: https://blogs.msdn.microsoft.com/appserviceteam/2016/05/16/disable-session-affinity-cookie-arr-cookie-for-azure-web-apps/
Original
I know that this is an old question, but it seems that the Azure IIS ARR has been updated to version 3 and among other new features it has "Session affinity opt-out".
Ben Ari has a post about it here: http://blogs.technet.com/b/erezs_iis_blog/archive/2013/09/16/new-features-in-arr-application-request-routing-3-0.aspx
The short version is that you can add Arr-Disable-Session-Affinity: True
to your response headers and sticky sessions will be disabled.
The easiest way is to add the header in the customHeaders
node of the web.config
file like so:
<system.webServer>
<httpProtocol>
<customHeaders>
<add name="Arr-Disable-Session-Affinity" value="True" />
</customHeaders>
</httpProtocol>
</system.webServer>