The feminine definite article does usually lenite, but there is a rule called the 'dental rule', 'homorganic rule' or 'sgian dubh rule'. When the word causing lenition ends in a dental (where the tongue touches (or nearly touches) the front teeth) then the following word does not lenite if it starts with another dental. When exactly this occurs is a bit vague as the rules are applied less than they used to be. Mark says in his dictionary (p.687)
The consonants d and t are not lenited after cha or bu and traditionally do not lenite each other, though this old rule is now not always strictly obeyed. Final n traditionally prevents lenition of a following d, t, l, n or s e.g. nighean donn bhòidheach a pretty brown-haired girl, though, again, this is not always strictly obeyed. For example, mòran daoine and is urrainn domh are now more commonly mòran dhaoine and is urrainn dhomh.
As far as an is concerned, that means that d and t are not lenited, and there are special rules for s anyway.
Note that he forgets to mention that dh and th also stop lenition as they used to be pronounced as in English this and thin respectively. You will soon (or already have) come across feumaidh tu where you might expect thu.
Mark does not explain that cha is on the list because it used to be chan and bu because it used to be buth.
d and t are shown in the notes as letters that do not lenite after the article, but it does not point out that this is contrary to the normal lenition rules, hence some possible confusion.
There is a very similar phenomenon in Welsh, as shown in the Welsh Duolingo notes The 1 where it is called 'weak soft mutation'. DD
Exactly. Words starting with t or d do not lenite after the definite article, and quite a few other words that end (or once ended) with d, t, dh, th, s or n. Historically it was to do with the tongue not having to move very far, but these days you have to learn the situations one by one.
A similar thing happens in Welsh, for anyone learning it, and they call it the week soft mutation. Again, you have to learn the details rather than applying general principles.