It is possible to do this; however, the assumption is that the path to the images is specified relative to the location of the properties file.
As an example, look at the CPA example here; the image path columns are specified as simply "images" which refers to the "image" folder. "images" is comntained in the same folder as the properties file, so if you move it, CPA will no longer be able to find it.
CellProfiler will always include the absolute path in the database, but the SQLite file can be modified after the fact. I've used the SQLite Manger extension in Firefox to change database contents; I'm sure there are others. For the example above, you can change the paths to whatever you need with the following statement query:
UPDATE per_image SET Image_PathNames_Path_OrigDNA = "<relative_path>"
where is the relative path to the images, in this case "images".