Is it advantageous to scale out using multiple SQL Server instances on the same server? For a use-case of 9 databases spread across 3 time zones, is it advantagous to spread the databases onto multiple SQL Server instances on the same server?

What will I gain and what will I lose?  Tagged sql-server-2012 because it's the most pertinent, although generally the question is across SQL Server versions.
No, no and again no. One instance, one server, Always. The reasons have to do with the SQL Server memory manager and CPU scheduler architecture, it really works best if it has the whole box to itself and nothing else runs on the box. 'Partitioning' of resources (max server memory, affinity mask) solves some problems and introduces more. 

If you want to consolidate, consolidate in one instance.

If you must keep separate instances then a far far far better option is Hyper-V and partition VMs. VMs have a much tighter control of resources and offer many other advantages. You need to be able to expose IO paths directly to the VM (no DBs on VHDs).