A person who doesn't have hair on their head doesn't feel good.

[tufguy]

People hate loosing hair. A person who doesn't have hair on their head doesn't feel good.

Do I need to use "does" or "do" After their in this sentence?

 

[probus]

Doesn't is correct because you used the singular noun person. Had you used people instead of person you would need the plural form "don't."

Loosing, however, is wrong. You mean losing.

 

[Annabel Lee]

Perhaps Tufguy got confused by the seeming plurality of gender-neutral singular "their" in the relative clause, which some might view as inconsistent with "a person." In the old days, before gender politics, the sentence could have been written like this, the masculine pronoun being the default:

A person who doesn't have hair on his head doesn't feel good.

 

[emsr2d2]

People hate loosing losing their hair. A person who doesn't have hair on their head doesn't feel good.

Do I need to use "does" or "do" after "their" in this sentence?

Note my corrections above.