Can I turn off the beeping from the monitored device when I access the online account?
You can easily run a test yourself when you have the phone in your hands. Simply login funamo.com and see if you hear a beep. I don't think that's the case. The only time when that may happen is when you do a push synchronization from the web. Then depending on the settings of the device, a notification may trigger a sound. But simply login online should not trigger a sound on the device.