To avoid ambiguity, you should use "my".
When I was 16 years old, the father said to me...
This is an odd construction. In some religious beliefs, it actually makes sense, though "father" would normally be capitalized:
When I was 16 years old, the Father said to me...
"the Father" can mean "God" and, in some Christian faiths, it is a term for a clergyman.
If you mean to say that it was your father, you must use "my".
You can also omit it entirely, though you would likely capitalize it here, too:
When I was 16 years old, Father said to me...
In this case, you're referring to him by his title (in this case, more of a name substitution), "father" and it's implied that you mean your own father.
As to the shirt example, it too can be ambiguous. If you want to imply that it's the shirt you are currently wearing which tore, it is more clear if you say "my shirt". The verb here should be "tore", the past tense of "tear".
As I fell, my shirt tore apart.
It is certainly possible to use "the shirt". For example, let's imagine you're in a fight with a sibling over a shirt. You each have one of the arms of the shirt and you're trying to get it away from your sibling. In this fight, you fall and as you fall, the shirt is pulled so strongly that it rips. In this case, you would certainly say:
As I fell, the shirt tore apart.