Does this imply lovesickness or an actual heart disease?
Only a physical illness. “He is lovesick” would be Tá sé i bpian an ghrá (“He’s in the pain of love”).
Go raibh maith agat. :)
Is there an Irish equivalent to the English language concept of "heartsick"?
See its entry in the EID.
When is tinn used instead of breoite, or are they interchangeable?
I would only ever say, "He has a bad heart." Possibly, "His heart is bad." I've never heard an adult say that someone's heart is sick and mean a physical illness.