I've had this question before. Here is an answer I found online posted by jaybird3rd:
There are a few different types of screwless cartridges. The INTV-era cartridges I own all have a square panel on the underside, and these look pretty much the same as the originals except that the panels are attached with plastic posts instead of screws (they might also have used some kind of adhesive). I don't know of any easy way to get these open without damaging the shell. The least-destructive way might be to use a 1/4" drill bit on the plastic posts (where the screw holes used to be; you can usually see the circle footprints), just enough to drill through the panel and separate it from the posts.
The "solid" shells (usually late Mattel and Intellivision Inc. releases) used a small plastic guide to secure the board. There is a pair of ridges molded into each of the side walls on the inside, and the cartridge board slides in between them. The plastic guide is then slipped in over the board fingers, and it has a pair of plastic tabs on the sides which snap into the ridges and hold the board in place. If your cartridge is one of these, you should be able to see the tabs if you look on either side of the board