Running .NET 1.1 Application on .NET 3.5

1. Is there any benefit of converting the .NET 1.1 application to .NET 3.5
Hopefully VisualStudio will do most of this conversion for you so it isn't that hard.
2. Is there any benefit in performance of application
You hope that your application will run faster? Likely not because it strongly depends on the application itself and how it depends on other elements like a database connection. When a lot of processing is done you could gain performance by using new available features that might help with that but it would be up to you to implement it.
3. Is it going to benefit me, as Microsoft has stopped support for .NET 1.1 or any other benefits.
A strong argument would be that it Enhances Security. Not only is .net 2.0 (and up) more secure but because it has support, new problems are fixed.

So in the long run it could be good to upgrade your .net version, also because it could get harder when new functionality is added. This would mean more conversion and maybe more work. You could also benefit from new features when adding new functionality. If it is a standalone application running for years without any trouble and no further future changes are expected, I would recommend to let it be for now.

Good luck!

.NET 1.1 is not officially supported any more. So it won't get updates or security patches. And a future OS may even be incompatible with it (as it had some legaxy API dependencies). The CLR has also considerably improved in efficiency and performance across the .NET versions. So potentially, you "may" see a speed improvement. You will also be able to take advantage of all the 100s of newer classes and features that have been added to .NET.

And as an aside, while many people continue to use remoting and insist it's better performing that WCF, remoting is "officially" obsolete. Microsoft recommends that you move up to and use WCF. So from the "official" Microsoft perspective, using remoting is kinda similar to using classic VB6 or Frontpage.