RubyMine doesn't have a feature exactly like Sublime Text file previewing.
The Quick Definition action can sometimes meet the same need. When invoked on a file, it opens a popup with the file's contents. You can invoke it on a file in the Project or Find pane and probably other panes, dialogs, popups, etc. that list files. (You can also invoke it in an editor on an identifier such as a class, module, or variable; it will show the code that defines the identifier.) Invoke Quick Definition with ⌥-space or ⌘Y in the Mac OS X 10.5+ keymap; use ^⇧I on Windows or check Preferences → Keymap for other keymaps, or use ⇧⌘A qd ⏎. Hit Escape or click outside the popup to close it.
When you invoke Quick Definition on a file in the Project pane and navigate to other files in that pane (with down and up arrows to move between files, and right and left arrows or Enter to open and close directories), the Quick Definition popup will stay open and update to the currently selected file, which is relatively close to Sublime Text file previewing. You can also hit Enter to open a file, F4 to open a file and move focus to the editor, edit the file, ⌘1 to jump back to the Project pane, etc., all with the Quick Definition popup still open, until you choose to close it.
If that isn't close enough, there are two feature requests for Sublime Text-like file previewing in JetBrains IDEs which you can vote for:
