Is it obligatory to observe the sukoon (vowellessness) when pausing during the recitation of Al-Fatiha, the surah, and adhkar in prayer, or does that fall under the rules of Tajweed?

Observing the silent stop (sukun) when reciting the Quran is obligatory by the consensus of reciters. Sukun is the original form of stopping on a word that is vowelled when connected (waslan), and it entails stripping the letter of the three vowels. Some have occasionally permitted stopping with rum (a slight vocalization) or ishmam (lip-pointing). However, not observing the silent stop (sukun) in the adhkar (remembrances) during prayer is a mistake (lahn).