"an t-sùil gheal"
Translation:the white eye
They are silent. This is a type of lenition of s. In the Old Irish the article was in, ind (from older *sindos, *sinda, etc.), ended in d. When it lenited the s of the following word, the s changed to /h/ and then the -nd h changed to -nt (the d of the article devoiced to t under the influence of h), thus in Old Irish the eye was int ṡúil pronounced /intuːl’/.
This t was later reanalyzed as a part of the noun, hence the modern spelling an t-sùil with pronunciation /ən tuːl’/.
More etymological spelling would be (but nobody writes like that) something like ant shùil.
You can read more about it on Gramadach na Gaeilge (website about Irish grammar, but the history here for Scottish is the same).
Of course they are treated differently than those starting with a vowel – nouns starting with an s do not start with a vowel.
A few comments above I explained that an t-s is just an irregular pattern of lenition of s after the article (and thus you can expect an t-s everywhere where you’d eexpect a’ + lenition with other consonants) and where it comes from.