For example in a tree, the water is lifted from the high concentration in the soil to the low concentration higher up in the tree. But at the end of that process the water has been elevated, which should take energy (=mgh), but it seems like it kind of gets lifted for free without spending any energy?
Similarly, dipping a paper towel into a bowl of water, the water “climbs” the towel (by capillary action?) and absorbs upwards, meaning the water was lifted upwards (so gained potential energy) seemingly for free?
Generally, yes, but in the case of trees there is also negative pressure (vacuum) exerted from transpiration when water leaves the top of the tree and “pulls” other water up behind it.
So the energy comes from sunlight and ambient heat, which provides the energy to evaporate that water, overcoming its adhesion and cohesion.
I think it’s more like sunlight energy + adhesion energy - cohesion energy, because the capillary action is also helping to lift the water but generally yes the energy is coming from a combination of the chemical forces in the water and the pressure gradient from the sunlight/heat.