"There is no place like 127.0.0.1" the old saying goes. While the loopback interface is most often seen as the interface where the 127.0.0.1 address is assigned by default and where the 127.0.0.0/8 network is routed, and just a way for programs on the same host to communicate over the network without actual network, it has uses in networked context as well.
Before we talk about those use cases, we need to discuss interfaces themselves. In some OSes, such as Cisco IOS, and many BSD derivatives, it is possible to create multiple loopbacks. Linux kernel (and thus VyOS) historically allowed only one loopback (named "lo"), and this behaviour has become too traditional and relied upon to change overnight, so to implement multiple loopback, a new interface type called "dummy" was added. Dummy interfaces are functionally identical to loopbacks so the difference is mostly aesthetic.
This is how to setup a dummy interface: "set interfaces dummy dum0 address ...". If your problem does not require independent interfaces, you can also just add another address to the loopback.
So, why would one want to use a loopback/dummy interface instead of assigning another address to a physical NIC?
Case 1: tunnel endpoints
We have already talked about GRE/IPsec behind NAT and/or with dynamic addresses. Since GRE requires fixed local and remote endpoint addresses to work, and in a setup where dynamic addresses or NAT is involved you do not have fixed addresses, the trick is to use a pair of addresses made up specially for this purpose as GRE endpoints.
Case 2: management addresses
Suppose you have a router A with two NICs, connected to networks B and C that are connected to each other, so that if any of the links fails, the network as a whole is still operational. However, if you choose either NIC A or NIC B address as a management address, it may become inaccessible if one of the NICs fail, forcing you to manually fall back to the other address.
To prevent this situations, people often assign a dedicated management address to a loopback, create a DNS record for it, and advertise that address to all other routers so that as long as there is at least one path to that router that works, they do not need to worry about addresses of physical NICs to SSH to the router, and are free to change those addresses without having to update the DNS or memorize the new address.
Case 3: iBGP peer addresses
Since iBGP uses the same autonomous system number for all routers, it loses the ability to use AS path for path selection and loop detection. This means to keep the network loop-free, one has to setup it as a full mesh, or use a route reflector.
If we use addresses of physical NICs for session endpoints, we run into the same problem as in the previous use case: a session goes down with the link even if there are other valid paths. A possible solution is to select dedicated addresses for iBGP sessions, assign them to loopbacks, and advertise them to all other routers through a link-state protocol such as OSPF.
Your use case?
If you know other cases when a network setup can be improved by using loopback or dummy interfaces, let us know!