Yes it is.
Trust me on this.
It's just that you don't understand what is happening.
C# code is executed on the server, not the client. So when you execute
You are executing it on the server, so the server is where the instance of internet explorer is run - not the client. When you were in development, both the server and the client were the same machine, so it looked like it worked - but when you deployed to the web server, suddenly you can't see Internet Explorer open because it is open on the server (which may be round the other side of the world) and is probably annoying the heck out of the Web server administration guy. Assuming the server has IE installed which is not a given - it is very unlikely that anything unnecessary is installed and that would exclude IE.
You cannot do what you want, except under very specific circumstances. Security will not allow you to execute applications (and certianly not specific applications which may or may not be installed) on a cline machine from the server. And the specific circumstances do not occur in 99% of circumstances.