The custom is (for Ashkanazim) that a shecheyanu is not made during the three weeks. See Siman 223 Sif 3 where you see that for "celim chadashim" (new vessels) a shecheyanu is made; in practice, though, people don't make a shecheyanu on furniture nowadays. Then would it be a problem to buy new furniture for ones house? For example, a large seforim shelf or any other large piece of furniture for the house that one would probably have simcha (joy) from having.
For Ashkenazim, it seems that an individual cannot buy furniture during the Three Weeks but a couple or a family can:
Sefardim disagree and do not allow it: