I've got a similar Azure/DNS question - I have the Site-to-Site VPN set up between Azure and the local network, have changed the DNS providers for the vNIC on the VMs to point to internal DNS numbers, have verified that vNIC has static IP and that it's gatewayed out to the VNet setup with the VPN connection. Machines on either side of the tunnel can ping each other all day long, no drops. But no DNS either.
I'll be happy to provide more detailed info, I'm posting a snippet of the IPCONFIG /ALL command here.
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C:\windows\system32>ipconfig /all
Windows IP Configuration
Host Name . . . . . . . . . . . . : XXX-YYYY-ZZZZZ
Primary Dns Suffix . . . . . . . :
Node Type . . . . . . . . . . . . : Hybrid
IP Routing Enabled. . . . . . . . : No
WINS Proxy Enabled. . . . . . . . : No
DNS Suffix Search List. . . . . . : reddog.microsoft.com
Ethernet adapter Ethernet:
Connection-specific DNS Suffix . : reddog.microsoft.com
Description . . . . . . . . . . . : Microsoft Hyper-V Network Adapter
Physical Address. . . . . . . . . : 11-22-33-44-55-66
DHCP Enabled. . . . . . . . . . . : Yes
Autoconfiguration Enabled . . . . : Yes
Link-local IPv6 Address . . . . . : fe80::99c9:d2e8:e22b:99f4%4(Preferred)
IPv4 Address. . . . . . . . . . . : 10.1.0.6(Preferred)
Subnet Mask . . . . . . . . . . . : 255.255.255.0
Lease Obtained. . . . . . . . . . : Friday, September 6, 2019 7:12:53 PM
Lease Expires . . . . . . . . . . : Tuesday, October 14, 2155 4:31:09 AM
Default Gateway . . . . . . . . . : 10.1.0.1
DHCP Server . . . . . . . . . . . : 168.xx.xxx.xx
DHCPv6 IAID . . . . . . . . . . . : 12345124
DHCPv6 Client DUID. . . . . . . . : 11-22-33-44-55-66-77-88-99-aa-bb-cc-dd-ee
DNS Servers . . . . . . . . . . . : 192.168.0.xxx
192.168.0.xxx
192.168.20.xxx
NetBIOS over Tcpip. . . . . . . . : Enabled
Tunnel adapter isatap.reddog.microsoft.com:
Media State . . . . . . . . . . . : Media disconnected
Connection-specific DNS Suffix . : reddog.microsoft.com
Description . . . . . . . . . . . : Microsoft ISATAP Adapter
Physical Address. . . . . . . . . : 00-00-00-00-00-00-00-E0
DHCP Enabled. . . . . . . . . . . : No
Autoconfiguration Enabled . . . . : Yes
Tunnel adapter Teredo Tunneling Pseudo-Interface:
Media State . . . . . . . . . . . : Media disconnected
Connection-specific DNS Suffix . :
Description . . . . . . . . . . . : Teredo Tunneling Pseudo-Interface
Physical Address. . . . . . . . . : 00-00-00-00-00-00-00-E0
DHCP Enabled. . . . . . . . . . . : No
Autoconfiguration Enabled . . . . : Yes