It depends on which version of the program you are using.
Automatic rendering was added as a feature a few versions back, but was removed because it cause the very issues you're describing.
What version of Premiere Elements are you using and on what computer operating system is it running?
The rendering of the Timeline content in the Edit Mode is a preview thing. When you see an orange (Premiere Elements 11) or red (versions earlier than 11) line over the Timeline content, that is the program telling you that you need to render the Timeline in order to get the best possible preview as seen in the Edit Mode Monitor. Being a preview feature, you do not have to do this type of rendering, but it is your window of opportunity to catch a problem sooner than later and correct it when still in the Edit Mode. It is especially useful for titles, transitions, effects, and non native video formats.
Keep in mind that rendering has two types...rendering for preview and rendering with regard to export encoding.
If you are troubled by rendered Timeline content (green line of Timeline content) in a project presenting as unrendered (orange or red line over that Timeline content) when you open that project after it has been saved and closed, that is an old story. It seems to happen more frequently in earlier versions but still can be encountered in later versions from time to time. Key thoughts:
How you define "unrendered" here is interesting. If you define "unrendered" based on the color indicators alone, then one might be dealing with "unrendered". The Render process is associated with the program automatically generating preview files and placing them in the Adobe Folder. In this instance of what is being called "unrendered" after re-opening the program, the preview files from the Timeline content render before save/close are still alive and well in the Adobe Folder.
In such cases, I have always recommended:
a. If you think that you require a preview, render selectively when you see orange or green over Timeline content, setting the gray tabs of the Work Area Bar to designate just that area that you want rendered.
b. move on.
I have seen no explanation for what gives the appearance of a disconnect between the preview files in the Adobe Folder and the color indicator system of the Timeline.
More information on this if necessary.