Because MOBO is usually cheaper than a CPU, try to replace the MOBO first. Also a bad PSU can destroy everything, use a better one instead.
That's debatable - I've had some cheap PSUs blow up and short out on me but I've never had any additional equipment taken down with it.
To answer the original question - no - I've moved CPUs from multiple motherboards before and been fine. Just be careful when you're installing the fan and you're good.