In the Human Japanese app, it says that chi and tsu become ji and zu respectively, while on Wikipedia it says that most commonly, they become shi and su.
Imagine the dakuten as a mark that transforms unvoiced phonemes (sounds that make up a language) to voiced phonemes. A quick-and-dirty way to tell whether a phoneme is voiced is checking whether making that sound makes you use your vocal cords or not. For example か (ka) is unvoiced, が (ga) is voiced. と (to) is unvoiced, ど (do) is voiced. You're making the same formation in your mouth to produce the sound, but the voiced one uses your vocal cords. This should hold true for EVERY pair of characters with and without dakuten.
So it holds true to adding dakuten to ち (chi) and つ (tsu). The issue arises because the characters resulting from adding a dakuten to "chi" and "tsu," (ぢ and づ respectively) are much more difficult to approximate with English phonemes. Try saying ち or つ while using your vocal cords and you'll get something like "dji" and "dzu." Much harder to romanize than "ka ki ku ke ko." While most Japanese phonemes have pretty straightforward English analogues, ぢ and づ are oddballs, backing up the age-old adage that learning the script of the language is immensely helpful to capture things that you might not catch with romanization.
As a side note, words containing づ are not super common (though they do exist, such as 続く／つづく and 葉月/はづき), and words with ぢ are even rarer.