As the size of a phonemic inventory decreases, the information rate allowed by the inventory should likewise decrease. So are there any (semantico-)pragmatic or morphosyntactic strategies that languages with fewer sounds might adopt to compensate for this inefficiency? Or is the effect negligible?
I recall a recent study at the Université de Lyon. It shows how, in order to maintain a roughly fixed information rate, languages make tradeoffs in any of a number of areas—but predominantly in syllable structure. Essentially, phonemes can be readily omitted in fast speech to increase information density, but syllables are dropped much more rarely.
The study compares English, French, German, Italian, Japanese, Mandarin, and Spanish, and uses Vietnamese as a point of reference for normalisation purposes. The two tonal languages (Mandarin and Vietnamese) were found to have the highest relative information density.
They also found that information density is not strictly fixed—there are outliers. In particular, Japanese has a small phonemic inventory and very simple syllable structure. So even though the syllable rate of Japanese (7.84±0.09 Hz) was comparable to that of Spanish (7.82±0.16 Hz) and French (7.18±0.12 Hz), Japanese was given an information density of 0.49±0.02—next to a more typical density such as 0.91±0.04 in the case of English.
So the effect is by no means negligible; anecdotally, English subtitles for Japanese dialogue go by noticeably slowlier than English subtitles for dialogue in other languages. Bugs me to no end. ;)
Put simply, they become more verbose, Hawaiian is a good example, where the words can be quite long, even nouns, e.g. Kamehameha. 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 to 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 contrast; the Pirahã 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 (e.g., tense vs. lax, length, and diphthongs).