Learn Portuguese (Portugal): O coração dói.

Great catch! Most words ending in -ção (like a informação) are feminine. However, coração (heart) is one of the few exceptions. It is a masculine word, so it takes the masculine article o.

What is the base verb for dói, and how does it work?

dói comes from the irregular verb doer (to hurt or ache). Because coração is a single thing (an 'it'), we use the singular form dói. If multiple things were hurting, like your eyes, you would use the plural form doem.

How would I use the 'ter de' (have to) structure from this lesson with this sentence?

You can connect this sentence to an obligation by saying O coração dói, tenho de ir ao médico (The heart hurts, I have to go to the doctor). Here, tenho de means 'I have to' and is followed by the infinitive verb ir (to go).

In English we usually say 'My heart hurts'. Why doesn't this sentence use a word for 'my'?

In Portuguese, when you talk about your own body parts, you usually just use the definite article (o or a) instead of a possessive word like meu (my). The context makes it obvious that you are talking about your own body.