Q: When I move a virtual machine to Azure, can it keep its existing IP address?
A: Virtual machines in Azure have a single virtual network adapter that must have its IP address assigned by the Azure fabric by configuring the OS within the virtual machine to use DHCP. The IP address assigned will be from the virtual subnets defined in the virtual network that the virtual machine belongs to; if no virtual network is defined, Azure will automatically choose a virtual subnet to use. It's possible to ensure that a virtual machine always receives the same IP address even if the virtual machine is deprovisioned (shut down from the portal) by reserving an IP address in the Azure fabric. This is explained in the FAQ "Set Azure VM Static IP Address." This lets you ensure that a virtual machine will always have the same IP address in Azure, very similar to configuring a static IP address within the virtual machine.
The question was whether a virtual machine can keep the same IP address when moving to Azure. The answer here is most likely no. Consider that the IP address is from a subnet; to be able to route between Azure and on-premises, you need to have different subnets defined so IP routing can be performed. In addition, in Azure there can't be overlap between the IP range defined in the virtual network and the IP ranges defined for local, on-premises networks. The only way to retain the existing IP address of a virtual machine would be if all the virtual machines in a subnet were moved to Azure and the subnet were moved to the Azure virtual network and removed from on-premises. But in reality, and given the complexities of routing, that's unlikely to be possible.
One possible solution in the future will be when and if Azure supports network virtualization between on-premises and Azure, which would allow abstracted networks to be defined completely separate from the underlying network fabric. However, Azure already uses network virtualization for its IP management, which means adding this support between Azure and on-premises would effectively be bridging the different virtual networks—which would require tighter integration for the management and control planes of network virtualization and certainly System Center Virtual Machine Manager integration.