If you are using Azure custom DNS servers (eg VMs configured in your VNET DNS settings) or Azure DNS resolver you can also configure conditional forwarding for these FQDNs from local/on-prem to your Azure custom DNS servers and your Azure custom DNS servers should forward them to Azure provided DNS (168.63.129.16). Also your private DNS zones in Azure should be linked to VNET where DNS servers reside and all other VNETS (if any) should be peered for proper DNS resolution.

I've encountered this issue too, shame that MS docs doesn't provide this information on AMPLS configuration guide (api.loganalytics.io and api.applicationinsights.azure.com are not mentioned at all).