This 「は」particle is sometimes called a "topic" particle. It emphasizes the topic and implies that there is more to the story. Please see the following sentences.

（１）机の上に日本語の本があります。 （２）机の上には日本語の本があります。

(1) means "There is a Japanese on the desk." (2) means the same. However, (2) implies that there might be something (a vase, for example) on a book shelf, for example, or that there is a bag under the desk. So, in (2), "on the desk（机の上）" is the focus (topic) of the discussion.

So, 「彼のしつこさには閉口する」implies that his persistence is particularly annoying among a group of friends (or co-workers or whoever might be compared to him).

I hope this helps.

You can say just '彼のしつこさに閉口する。' without '-は'. '-は' just emphasises it. 閉口する takes '-に' to show what causes your annoyance. In english you may say the example sentence as 'His importunity makes me sick.', so I guess you may have thought 'his importunity' as the subject of this sentence but it is not in Japanese. In this Japanese example sentence the subject should be 'myself' but is omitted as usual way.