Put simply, they become more verbose, HawaiianHawaiian is a good example, where the words can be quite long, even nouns E, e.g. - KamehamehaKamehameha. When human languages lose complexity in one area they make up for it in another.
A strategy when a language loses case endings, possibly due to phoneme merger & loss, is that this enforcesto enforce stricter word order. As without the erstwhile case-endings the Subject(Topic) and the Object or Verb(Comment) cannot be easily differentiated.
Another strategy has been the use of tones for phonemic contrastcontrast; the Pirahã languagePirahã language is a good example of this.
Note: You specifically mentioned minimal phonemic inventory, in case of a smaller consonantal inventory, one of the strategies used is to increase the number of vowels (Ee.g. -, tense vs. lax, length & diphtongs, and diphthongs).