This mostly depends on what you mean by "unlimited". For all practical purposes, solar power is unlimited. You can use it, for free, for as long as you wish. This is only unlimited over time - you can't have it all right now. If you mean what if everyone suddenly had access to as much energy as they like all at once, then it depends on whether they use it. If they do, then the answer is that the earth would heat up incredibly rapidly and possibly wipe out the population before they realised that they were causing the problem - before they had a chance to switch their energy sources back off again. Only using what you need would cause much less warming. Some people would know in advance not to use unlimited amounts of energy all at once, but those that don't, even if a tiny minority (which is optimistic) would have unlimited energy, so even a small number of them could use their energy to fry the earth.
Obviously neither of these extremes gives a meaningful answer. In order to get specific answers you would need to specify how much energy per second each person has access to.
A few solar panels each
If everyone has the equivalent of the roof of their house covered in solar panels, then everyone can save money by not having to pay energy bills, but no one gets much more benefit than that because you can't sell energy if everyone has more than they need already. So the effect would be fairly small. People still need to pay for other things, so people still need to work.
A hydroelectric dam each
The Three Gorges Dam produces 22.5 gigawatts, enough to power over 18 flux capacitors. If everyone has the equivalent of a large hydroelectric dam then inevitably people will start to use more than they did when it cost them in bills. Exercise rates will drop in people who are not interested in exercise, leading to a greater divide between people who are fit and healthy and people who are not. Temperatures in cities will become considerably higher than in rural areas. At this level people are unlikely to be killed off by a sudden rise in heat, so they would have time to realise that there is a connection and would limit themselves to a tolerable temperature. It would still be uncomfortable though (if it's comfortable then why shouldn't I use a little more energy - the people next door will be...). This may lead to increased migration from cities to rural areas, gradually redistributing the population. This would in turn allow more energy to be used overall without making local temperatures intolerable. Therefore the overall global energy use would continue to increase and the warming effect on the earth as a whole would increase. Without regulation, the temperature would continue to gradually increase as people spread out more and more, with likely increased extinction rates. Eventually food production would suffer and at some point regulation would become unavoidable. How long it would take before regulation was enforced and how long it would take to recover from the damage can only be guessed at.
