What kind of hardware problems could happen with my computer?
That's only if your hardware already has problem in the first place.
Otherwise, you'll just experience high CPU, HDD, RAM and Network activity while Bitcoin Core is running.
It's common for a computer to run normally if it's idle or just at minimal usage, however if it's pushed to the limit, existing hardware issues may show-up.
The most common "
exposed problem" when syncing Bitcoin Core are: HDD or RAM issues that could corrupt the blockchain.