All breaking errors should raise an Exception which will be displayed in the Icinga Director Web UI. Make shure, you check the configured importers by looking into the Director / import source, select the import source to test and click check for changes and trigger import run manually. If there is no error message, this is a good sign.
You also can make use of the preview tab in the importer setup web page in the Icinga Web2 UI right where you manage or create the importers.
For extended trouble shooting, the Azure API pluging makes use of the Icingaweb2 standard logging function. Depending on your Icingaweb2 configuration this might be a log file or routing log messages to your syslog facility.
The Icinga Web2 Azure API module logs a good bunch of information level messages into syslog. This will enable you to nail down where the import run breaks if it does. There is, of course, also some more logging, if switching log level to debug. But don't expect to much as I only add debug log level messages for killing bugs or testing. So not any piece of code is covered with debug level logging stuff.
Additionally, the importer will log warnings and errors with these log levels set correctly if encountered. Make shure not to mask these in your syslog service.
The somewhat verbose information level log entries can be ignored safely in normal operation mode. Same goes to debug log level.