We haven't done many systematic performance comparisons. However, it seems to be the case that whilst Mono still performs worse than .NET in a few areas (e.g. loading new AppDomains), in general there isn't a significant performance difference between running OpenSimulator on Mono and on Windows.
.NET is not a language, presumably you're referring to C# vs. Java? C# may have the upper hand over Java currently, but Java 8 is a very good sign from Oracle that they're finally willing to push the language forward, which down the road may not necessarily mean C# > Java.
As for .NET being so much better than the JVM, that's not a factual statement ;-) Certainly the JVM, with Clojure, Scala, Kotlin, Ceylon, Groovy, JRuby, etc. running on it, has proven the JVM to be a pretty awesome environment for emerging languages, while the CLR has proven to, not surprisngly, be an awesome environment for Microsoft backed projects.
If you're tied to Linux and Mono isn't a viable solution, why not try one of the above Java alternates?? JVM is a kickass dev environment these days if you're able to break away from Java...
FWIW, last time I tried Mono (a couple of years ago) it was quite impressive, particularly the IDE, incredibly responsive. The biggest drawback was the feature lag between latest and greatest from .NET side of the fence.