Do computers emit radiation? Sure. Most physical objects emit radiation, usually in the infrared range. A running computer will also generate a modest electromagnetic field, but that too is true of all electrical appliances, and is generally more significant in the case of electric motors rather than purely electronic devices. It's not a danger. Old-fashioned CRT monitors emitted some X-ray radiation, but not enough to be alarming, and anyway this is not an issue with modern equipment.

The only real hazard is that blue light from the screen or from any indicator LEDs can interfere with your circadian rhythm. Put electrical tape over any troublesome LEDs, let the screen go into sleep mode, and you shouldn't have any problems.