While TextMate doesn't have built-in FTP or SSH capabilities there are a few bundles for it that have tried to remedy this shortcoming.
In particular the FTP/SSH Bundle aims to provide what you seek: a way to define a "project" that can be synchronized (both up and down) with a remote file system. The bundle is a touch dangerous if you're using FTP as it requires your password be left in plaintext in the
.ftpssh_settings file that you leave on disk at the root of your project folder to tell the plugin how to connect to your remote file system.
The other option, if you're not set on TextMate and just want something like TextMate, is to take a look at Sublime Text 2. I'll be honest, I was a die hard TextMate guy, but in the last six weeks Sublime Text 2 has won me over. The active development from the company on the product, the very active bundle community, and it's incredible speed when it comes to working on large files have really taken me in. There's an FTP, FTPS, SFTP plugin for ST2 that's a cut above the plugin for TM I mentioned above. It leverages keychain for password storage and doesn't require you keep your passwords in plaintext on disk for it to work. It also lets you set up mappings for remote <-> local connections directly from the editor UI.