A general word for weapons which are not considered firearms

Cold weapon (also known as white arm) seems to be an appropriate word.

It refers to weapons that do not use any sort of explosive force in their function. It comprises blunt (knuckles, maces), edged (swords, knives) and ranged (bows, crossbows) weaponry.

Here is the relevant excerpt from Wikipedia:

A cold weapon (or white arm) is a weapon that does not involve fire or explosions (such as the act of combustion) as a result from the use of gunpowder or other explosive materials. Ranged weapons that do not include gunpowder or explosive materials and melee weapons are cold weapons. Firearms and explosives (such as grenades, land mines, missiles, bombs, etc.), and other weapons which rely on heat or burning are not cold weapons.

"Knives, knuckles, a piece of metal, etc." can be considered melee weapons, although my impression is that the term is used more commonly in gaming (where it stands in opposition to ranged weapons) than in discussions of real weaponry.

In any case, there are lots of other kinds of weapons that are neither firearms nor melee weapons, such as bows and arrows, catapults, bombs, et cetera, so if you really want to include all possible non-firearm weapons then Josh61's suggestion may be your best bet.

They can be referred to as: non-firearm weapons:

Non‐firearm weapon use and injury severity: priorities for prevention.

• To test the hypothesis that weapon‐related violence (excluding firearms) results in more severe injury relative to the use of body parts (fists, feet and other body parts), and to rank order of injury severity by assault mechanism.

Source: http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2564418/