Most of my heirlooms were upgraded to 110. I have a new Kul Tiran druid, and I asked, which heirlooms should I take to 120? Armor upgrades were 5K gold, and weapon 7.5K gold to 120.
Weapon- yes. Even though you are given a 180 weapon, you see few until the end game at 120, so that upgrade is worth it. More DPS is a faster trip up.
Head, Shoulders, Chest- no. You will get 280 pieces here fairly quickly. One from the initial heart quest, and another from 100 war resources turn in. I would only do 2 at a minimum, and your chest gives the most and is likely your best choice to take your first pick on.
Cloak, Legs, Rings, Trinkets- yes. It is nice grinding this stuff up in the scrapper. There are fewer options for rings and trinket choices, and this means more alts will experience them.
Neck- no, you get the heart neck piece right from the start.