I would like to get a new computer, but I have a lot of software that I use which is discontinued and requires OS X 10.8. Is there a way on a modern mac to have a dual-boot situation so that I can have the latest OS, but also have the option to boot into 10.8 without any performance issues?
Dual booting is likely out, the older OS won't have the drivers for newer hardware. What you could do is run the older OS in a virtual machine. There's VMWare Fusion, Parallels, VirtualBox, and maybe more I'm missing.
I run virtual machines routinely and don't experience performance issues so long as there is enough RAM. As I plan to run virtual machines I buy computers with plenty of RAM from the start. After that the performance is limited like it was installed "on the metal". There's some quirks with virtual machines that may or may not be problematic and they generally have workarounds. One issue that comes up is the guest OS is effectively behind a NAT internet sharing from the host, if this is a problem then it can be rectified with a USB network adapter.