<p>I'm using shared ASP.NET hosting and I have no remote-desktop access to the web server.  I only have the ability to FTP ASP.NET related files to the server.</p>

<p>In this scenario, how would I see how much memory my ASP.NET application is consuming?</p>

Could you create a page that listed various performance counters in a password-protected part of the site?  That would be my suggestion, along with making sure that somewhere on the page, a machine name or IP is listed so that should the site move among various servers, this can be noted somewhere.  Another idea would be to either write to a log file or send an e-mail with various performance counters so that you could see a history of how the server is doing, but be careful on the e-mail that this could become spam-like if you do it too frequently.