The answer to this strongly depends on the type of material from which the container is made. This answer also applies to heating in the microwave.
Clay-based, steel (don't put this in the microwave), glass and similar containers are generally considered safe to heat and reuse. Cardboard and plastic-coated cardboards (these are the shiny cardboards, like the ones often associated with "Chinese take-out" in American movies) will soften and fall apart.
There are several different types of plastics that might possibly be used in food containers. One major concern is the softening of the plastics at temperatures below the cooking point. What temperature this happens at depends on exactly which plastic the container is made of and the formulation of the plastic therein. Some plastics such as polyvinyl chloride (PVC) and LD-PE (low density polyethylene) will soften as low as 60 C (140 F). Expanded polystyrene containers will start to fall apart at a fairly low temperature too, and the polystyrene itself can soften as low as 90 C (194 F) depending on formulation.
The second major concern with plastics is that they contain chemicals that are added to adjust their physical properties, such as higher/lower melting point, more/less brittle, more stable, etc. Many of these chemicals are considered to be dangerous to human health. These can include things like Phthalates, which have been shown to have hormone mimicking properties, causing endocrine disruption and birth defects. BPA (bisphenol A) is another hormone mimic commonly found in plastics that is linked to all sorts of hormonal effects, particularly in young people.
In many cases these additives are completely stable (at least to my knowledge) in their initial plastic form, but once heated start to leach the additive in a non-stop fashion. This means that once heated (even by washing in hot water in some cases) the additive will start to leak out of the plastic into whatever is in the container, and won't stop doing this, so every time you eat from that container, you eat some of the additive. No-one fully knows what these additive effects are and how cumulative they might be, but the few scientists that I know personally, working in the area consider them to be a risk that is worth further study and recommend never heating plastic containers, or at least those known to contain BPAs, as do many authorities such as the Mayo Clinic, and BPAs have been banned in baby/infant bottles and similar items for a while now by most authorities such as the FDA and EFSA.
Edited to add: OP was asking about heat transfer in containers. Conductive containers such as metals will transfer heat faster than non-conductive containers, such as plastics, which are relatively impermeable to heat (especially the expanded foams, which are excellent insulators). The others are in the middle and depending on the thickness of the respective container might be faster or slower than plastic. Heating in the microwave is different to heating in a steamer, as what happens is that the microwaves interact with polar molecules in the substance (mostly water in food) and induce heating, a process known as dielectric heating. Plastics are mostly non-polar so don't interact strongly but also don't block microwaves much so heating is more or less unimpeded in plastic containers in the microwave. You can't use metal bowls in the microwave, but assuming no arcing (DON'T TRY IT; it's bad for your microwave and it can cause fires), metals will actually shield the food from the microwaves and prevent heating.
