I would not want to be inhaling vapors emitted by any of the parts of a laptop or desktop, if those parts were overheating to the point that they were giving them off or cracking apart. Not to mention, continuing to run said device is likely an incredible fire hazard.
If the device is still within warranty I'd contact the manufacturer and if not, I'd trash it and start looking for something new. In any case, no, I would not breathe in or allow anybody else to breathe in any offgassing from the materials used to build any computer system or device.