Do the total radiation emissions of manufactured devices affect climate change as much as solar radiation does? Do the cell phone emissions and the electric use of electric service providers worldwide, produce radiation with a climate change affect? Is this radiation comparable to the carbon dioxide impact on climate change?
TL;DR Neither the Sun's output nor radiation from electronics matters nearly as much for climate change as the increase in greenhouse gases.
First, to clarify what exactly we're talking about here: the relevant type of radiation when we're talking about climate change is infrared radiation, specifically, the particular bands of infrared radiation that are absorbed by carbon dioxide, water vapor, and other greenhouse gases (along with other bands that are absorbed by the earth itself). So the visible light that the Sun emits does not generally affect climate; similarly, the radio waves and visible light that most electronics emit by means of their operation don't do anything either.
So the question becomes: which produces more infrared radiation, the Sun or electronics? It turns out that about half of the Sun's energy is in the form of infrared radiation (source: a Wikipedia analysis of NREL data, https://en.wikipedia.org/wiki/Sunlight#cite_note-6), and the Earth receives, averaged over a day, 84 TW of power from the Sun (source: http://zebu.uoregon.edu/disted/ph162/l4.html), this means that the Earth receives roughly 40 TW of infrared power from the Sun. In contrast, the total amount of electricity generated on Earth is 23,816 TW-h/yr (source: https://en.wikipedia.org/wiki/World_energy_consumption), which works out to roughly 2 TW. But even then, most of the world's electricity consumption is not by electronics; in the US in 2013, only 4.5% of the total electricity consumption was by consumer electronics(https://www.cta.tech/CTA/media/policyImages/Energy-Consumption-of-Consumer-Electronics.pdf), and in developing countries, this percentage is likely even lower, so it's safe to say that the total amount of electric power consumed by consumer electronics worldwide is less than 100 GW (0.1 TW). If you add in things like servers, datacenters, and other commercial electronics, they consume roughly 10% of the world's power, so this figure is bumped up to 300 GW (0.3 TW), which is still less than 1% of the total solar infrared incident power. So the Sun's infrared radiation is much more important in terms of the total amount of infrared radiation in the atmosphere.
But your question was: which one is more important for climate change? In that case, what matters isn't the absolute value of the amount of infrared radiation present, but rather, the change in that amount over time. The solar irradiance changes regularly due to things like sunspot cycles, which cause changes on the order of 0.1% of the total solar irradiance (about 40 GW) (https://arxiv.org/pdf/1412.3935.pdf). To get the same kind of variation from electronics, you would have to change the power consumption rate by a relatively drastic, but not impossible, 10%. So the natural variation in solar irradiance is actually somewhat comparable to a drastic shift in electronics energy consumption.
That said, the changes in the amount of infrared radiation present in the atmosphere in the last century are nowhere near as large as the change in the amount of greenhouse gases like CO2. Remember, taking away all of the electronics we currently have would only decrease the amount of infrared radiation in the atmosphere by less than 1%, while the total amount of yearly CO2 emissions has increased by 18,000% from 1900 to 2013, causing the atmospheric CO2 concentration to nearly double (https://ourworldindata.org/co2-and-other-greenhouse-gas-emissions)*. This change dwarfs the other two, both in its magnitude and in its consequences for runaway warming. If the electronics are not powered by renewable energy, then the fossil fuels consumed by their operation will add to CO2 emissions, so, even though the infrared radiation produced by electronics isn't a significant factor in warming, the CO2 emissions produced by them still warrant limiting your electricity consumption.
*You might notice that the amount of CO2 in the atmosphere has increased much less than the amount of CO2 emissions has. This is because our planet is remarkably good at absorbing CO2. The oceans dissolve a lot of it, which is the primary cause of ecosystem destruction via ocean acidification. A lot is also taken up by plants, algae, phytoplankton, and cyanobacteria. The fact that, despite all of the carbon sinks working to keep CO2 concentrations at equilibrium, CO2 levels are still sharply rising, should illustrate just how much we've unbalanced the system.
The Earth easily radiates back into space all radiation power absorbed from the Sun and also the in comparison tiny additional power released into the system by human activity (generated mainly in nuclear and conventional plants or engines and eventually converted to heat by lossy processes, as in car brakes). This lossy human activity by the way includes, as you mention, the - really small - contribution of energy that lived as an electromagnetic telecommunication signal before also ending up as heat. The total human activity component is only on the order of 0.012% of the total energy absorbed from the Sun. There is a roughly 4th-power law between the Earth temperature (seen from space) and the total power radiated into space, so switching off the heat generated by all human activity would reduce the Earth absolute temperature seen from space by a mere 0.012% / 4 = 0.003% or on the order of 0.01 kelvin. So this is not what we mean by global warming. Global warming arises from slightly changing the composition of the atmosphere, in a way that renders it more opaque to the infrared radiation the Earth uses to radiate solar energy back into space. This allows the Earth biosphere temperature to rise while the Earth temperature seen from space remains essentially unaltered (it still has the same value required to radiate back exactly all energy absorbed from the Sun, according to the 4th power law mentioned before). So the opaqueness works both ways, it traps heat at the Earth surface, and it maintains the original cool outside view that determines the unchanged Sun-Earth equilibrium.