There are no Smyths/Smiths from Teeshan in the Ahoghill Old Churchyard records I have. There are burials from Teeshan, but not for that family.
There are some Smyths from Teeshan with gravestones in Kirkinriola graveyard. Have you checked there?
Erected by John Smyth of Teeshan, In memory of his son JOHN SMYTH, who departed this life 10th october, 1863; aged 3 years. Also THOMAS SMYTH who departed this life 9th March 1872 Aged 10 years. And the above named John Smyth died 8th April 1896 aged 76 years. and their daughter Mary Ann, died 24th Dec 1900 aged 50 years. and their daughter Jane Smyth died 2nd Feb 1905 aged 56 years. [Mar]garet Jane daughter of above Jane Smyth died 6th March 1920 aged 44 years
ERECTED BY Nathaniel Smyth of Tyshan IN MEMORY OF JANE his beloved wife who departed this life 6th October 1852 aged 64 years.
Erected by John Smyth of Teeshan in memory of his beloved wife MARGARET SMYTH who died 12th February 1888, aged 60 years. Also their daughter Isabella Smyth; who died 29th Nov. 1885; aged 20 years And their daughter Martha Smyth; who died 29th Sep.1887 aged 36yrs.
Not sure what burial records they keep in the council offices. Might be worth checking with them in Ballymena.