For a developer with a Java background, I am interested in exploring software development using the ASP.NET tools/platform as well.
Java web applications (.jsp and servlets) can run on many server platforms.
Question: Will a .NET web application be able to run in a Linux based server? Considering the scenario of not being able to use a Windows server for hosting a web app.
Yes we can. get familiar with Mono Project and read this article to get started.
The Mono project is your best option. However, it has a lot of pitfalls (like incomplete API support in some areas), and it's legally gray (people like Richard Stallman have derided the use of Mono because of the possibility of Microsoft coming down on Mono by using its patent rights, but that's another story).
Anyway, Apache supports .NET/Mono through a module, but the last time I checked the version supplied with Debian, it gave Perl language support only; I can't say if it's changed since, perhaps someone else can correct me there.
dotnet is the official home of .NET on GitHub. It's a great starting point to find many .NET OSS projects from Microsoft and the community, including many that are part of the .NET Foundation.
This may be a great start to support Linux.
You might want to consider this guide that helps Windows developers port their code to Mono/Linux:
Guide: Porting ASP.NET Applications - Mono
I can speak from experience. Even if your ASP.net website only uses .NET libraries supported by Mono you are going to have a hard time getting it to run if its anything beyond Hello World.
You won't have to re-write much code but you will spend hours/days/weeks dealing with little issues with mod_mono/xsp/apache configuration and file permissions and error handling and all the little things that go into a large website. (Be prepared to spend a lot of time asking questions on serverfault :) )
The problem is that a lot of people don't use Mono for ASP.net websites and so there aren't as many people reporting bugs so a lot of things that are minor bugs go un-fixed for a long time.
Since the technologies evolve and this question is top ranked in google, we need to include beyond the mono the new asp.net core, which is a complete rewrite of the asp.net to run for production in Linux and Windows and for development for Linux, Windows and Mac: