Heat is there for your equipped armour and weapons to level up, by filling up the 'heat bar'. The heat bar fills, the more you acquire. Hence the name. If you pay close attention to the details on the bars of your equipped items, you'll see different coloured bars, at the end of these bars, is an unshaded bar. As the item levels up, the unshaded bar becomes more and more shaded; until the item reaches level 10. At level 10, the item cannot be levelled any higher and has reached its full potential for protection or damage.

At level 10, some items feature special bonus effects too, to add further incentive to level them.

When an item reaches level 10, and you'll have to buy a new version to level up, or upgrade the item via alchemy. I use the term 'upgrade' loosely here, as you don't necessarily upgrade it. Some alchemy recipes require armour/weapons at level 5 or above to complete, hence being like an 'upgrade'. An example is creating a "Firotech Alchemer Mk II" requires, amongst other materials, a "Firotech Alchemer". The Mk II version is the same, but stronger.

So, every level, the weapon IS getting stronger, you just haven't noticed yet ^^