I got a reply from Windows Azure Web Sites Team on MSDN forum:
We have identified this issue as a bug in new Auto-Heal feature we shipped recently. Please add following in your web.config file to mitigate this issue for now.
<configuration>
<system.webServer>
<monitoring>
<triggers>
<slowRequests timeTaken="02:00:00" count="1000000000" timeInterval="00:01:00" />
</triggers>
<actions value="LogEvent" />
</monitoring>
</system.webServer>
</configuration>
We are working on releasing the fix ASAP.
Apologies for inconvenience.
Windows Azure Web Sites Team
Update: The workaround works only in Standard mode. It throws an error in Shared or Free mode, so be aware.