Tine is the Irish word for fire.
"na Tine" means "of the fire" Sir isn't an Irish word.
The name Aiden means little fire in Irish (Aodhán). Otherwise 'tine bheag'.
tine
tine / dóiteán
tine lonrach
tine pronounced tin-e
In Irish it's "tine bheag"
Modern Irish - tine; Old Irish - teine Scottish Gaelic - teine Manx - aile The name Áed is another word for 'fire'. Delbaeth means 'fire shape'.
tine tempha
In Irish it's "bean na tine"
There is no such language as Celtic. The word "Celtic" refers to a group of more than a dozen different languages, 6 of which are still spoken today: Irish Gaelic Scottish Gaelic Welsh Breton Manx Cornish In Irish it's: tine (controlled fire) / dóiteán (blaze)
It's not an Irish name so it has no meaning in Irish.