Learn Russian: Почему они всегда таскают тяжёлую мебель по квартире? - Why do they always drag heavy furniture around the apartment?

Таскают comes from таскать, which is the multidirectional verb for dragging or pulling. It is required here for two reasons: the word всегда (always) shows this is a repeated habit, and по квартире (around the apartment) shows the motion is in multiple directions rather than a single trip from point A to point B.

Why does the adjective change to тяжёлую but the noun мебель doesn't seem to change in the accusative case?

Мебель is a feminine noun ending in a soft sign. These types of nouns do not change their form in the accusative case, so it stays мебель when being acted upon. However, its modifying adjective still has to take the standard feminine accusative ending, which is why тяжёлая becomes тяжёлую.

Why is квартира in the dative case (квартире) here?

We use the dative case after the preposition по when describing movement around, along, or all over a surface or space. Here, по квартире translates to moving the furniture around the apartment.

Does таскать always imply that the object is heavy, like furniture?

Usually, yes. It translates to dragging, hauling, or lugging something that takes physical effort. However, native speakers also use it colloquially to mean carrying something around constantly or pointlessly, even if it is light, like a child lugging a favorite toy everywhere they go.