This question might be slightly subjective, but I am unsure where else it would be better suited.
I have used Orchard on a number of projects where the server was Windows Server based - integrating with AD, SQL etc to provide a MVC based portal - like stealing candy from a toddler!
I have been exploring MonoTouch recently, and installed Orchard on OSX under ModMono as per http://docs.orchardproject.net/Documentation/Running-Orchard-on-Mono.
It all seems to run fine, and fast, but I am unsure whether to back this on a commercial venture. I have always used MySQL or MSSQL as the DB, whereas my current install is running PostgreSQL - something new (which is always good).
Does anyone know of any sites which use Orchard running on Mono, with ModMono (Apache2) and PostgreSQL?
I plan to deploy to a cloud server running CentOS and Mono to do further testing before going any further, but think it could be an exciting avenue to explore.
I can't say I know of any sites that run Orchard on Mono commercially but I do know of a lot of sites that use Mono, Postgresql and CentOS. But if it works on a Mac running Modmono and Postgresql then it should work on CentOS.
The only issues that you may face running on Linux is case sensitivity. If you do run in to case sensitivity issues when you are trying out Orchard on Linux then you may want to configure your site to use Mono IOMAP temporarily until you fix an issues with case sensitivity.
This is a very reliable and cost effective stack. It may take a bit of research and problem solving to get up and running initially but once it's running it should be very stable.
I am currently near the end of porting an application that is going to be using Mono, Postgresql and Ubuntu running on Amazon EC2. I believe that Ubuntu is better than CentOS when it comes to Mono as it is much easier to setup and seems to be better supported. You can use the badgerports repositories located here if you want to try out the latest version of Mono on Ubuntu.