Laser and other beamed energy weapons, in orbit or from Earth, could potentially make their targets malfunction from overheating and perhaps even partial melting. Would this be a way of space warfare that is safe from causing much hazardous debris in orbit, compared to kinetic anti satellite weapons? Or would for example fuel tanks and batteries explode anyway?
This would still cause debris, either immediately or eventually. Technically, an unresponsive satellite is 1 piece of space debris. But the real question is lots of small pieces....
Overheating could certainly cause the tanks to explode at the time. If left disabled, a satellite with non-empty tanks and charged batteries does pose a threat to explode/breakup. That's why the accepted decommissioning practice is to empty the tanks and drain the batteries. An Air Force Weather satellite recently broke up because of a battery explosion:
I would think that it would be very unlikely. The beam from an earth based system, even if very powerful would be large and spread out by the time it has reached the satellite especially if its in a high orbit like geostationary orbit. A space based system although potentially much closer to it's target would be much more power and heat limited than a ground based system. Satellites are already designed to last for many years bathing in the intense solar radiation of of space(~1.3 kw/m^2) without any damage.
A beam for example, 10 meters by to meters(I'll pretend the beam is square to simplify things) when it reaches the satellite would would have to be provided by a 130 kw laser to even equal the intensity of sunlight. So any weapon able to destroy the satellite would likely be megawatt level. Because of the difficulties of keeping a several megawatt laser(or many smaller lasers) focused on a very small area after thousands of kilometers, a more conservative and much less power hungry approach then destroying the satellite itself would probably be taken such as frying the satellites communication or camera systems, essentially making it useless.
If the laser were powerful enough to cause significant physical damage other than just frying sensitive components and sensors then explosion might actually be a risk. The biggest risk of a large explosion would be the fuel tanks especially if they contain mono-propellant. A tank rupture could be caused by it heating up and causing the walls to lose strength or by the boiling of propellant causing the tank to over-pressure. A tank rupture in and of itself poses a relatively low risk for producing debris as it is likely to just tear apart into only a few pieces with not a huge amount of energy.
If the tank is filled with a hypergolic propellant then as long as only one tank is ruptured at a time and the two components do not mix then the outcome shouldn't be much different from a normal over-pressure tank rupture. If the satellite uses mono-propellant, Then the outcome could be much worse likely resulting in an ignition of the mono-propellant and a very large explosion. Batteries could also pose an explosion risk, however this would be relatively low energy and would require the battery be inside a pressure vessel.
Keep in mind that the majority of the satellite would be covered in reflective foil and/or polished which would make it much harder to damage the internal systems.