Q. Can I use an Azure load balancer to balance traffic between Azure and on-premises?

A. No. The software load balancer in Azure is used to balance traffic between Azure VMs only and more specifically Azure VMs in the same availability set (which means also within the same Azure region). If you wish to balance traffic between different regions/on-premises typically the architecture would have a load balancer solution at each location balancing traffic between service instances and then a DNS balancer globally which would redirect requests to one of the load balancers. Azure Traffic Manager is an example of this type of solution and can have endpoints which include Azure services and on-premises services.

Each Azure Traffic Manager instance has a unique DNS name (which can be aliased with a name from the organizations own domain) and when a request is made it is resolved to one of the endpoints based on factors including the latency of the request to the possible endpoints, round-robin, failover and geography.

Another option would be the Azure Application Gateway solution which is a layer 7 solution that can front services regardless of the region/on-premises. All that is required is communication from a network perspective.