If we are to focus on very old formations, (Beekes 2011, 13.1.2) discusses substantive derivatives of PIE suffixes in -r-, which have indeed yielded many original Latin substantives in -r:
We shall take the forms in -r and -n as examples to discuss.
1 -er-‚ is used for the nominative of the neuter r/n-stems: ∗iekʷ-r ‘liver’; ∗h₁ésh₂-r ‘blood’: Skt. ásr̥-k, Hitt. eshar, Gr. éar. Adverbs in -er: ∗(s)uper ‘above’: Skt. upári, Gr. húper, Lat. super.
 -ro-‚ we find in a number of old words: ∗h₂eǵro-‚ ‘acre’; ∗uiHro-‚ ‘man’: Skt. vīrá-, Lith. výras, Lat. vir, OIr. fer, Goth. wair [...].
 -ro-‚ also forms primary adjectives: ∗h₂ugró-‚ ‘strong’: Skt. ugráâ-; ∗h₁rudhró-‚ ‘red’.
 -ter-‚ in words indicating kinship: ∗ph₂tá-r, and nomina agentis: ∗ǵenh₁-tōr ‘conceiver, father’: Skt. janitár-.
 -tro-‚ is used for neuters which indicate the instrument: *h₂(e)rh₃trom ‘plough’: Gr. árotron.
 -tero-: see the comparative[...]. Compare also *kʷoteros ‘who of two’
(enumeration is mine -kkm). So this is a very good clue where to look for the substantives in Latin with the inherited -r. Under the same categories, the examples are:
- Often body parts are in this inherited neuters category: cor, venter, iecur, uber, uter, viscer, femur; emotional states: sopor, timor, torpor, pauor, moeror; abstracts of quality vigor, squalor; names of perceptibles: aural clamor, squalor, nurmur, fragor, fremor, visual nitor, fulgor, olfactory odor, fetor, tactile fervor, caldor, tepor, algor.
- Also ager, arbor. Seasons and parts of day: ver¹, vesper. There are very old tree names that probably belong to this category: acer, robor, etc. Meteorological: fulgur, imber.
- rubor, niger, ater. There are many primary adjectives in -r in Latin in general, but the question, as I understand, is focused more on nouns.
- pater, mater, frater²; cannot think of more. Both PIE and Latin are rich in kinship terms, but they did not end up ending in -r in Latin (as sobrinus/sobrina or socrus—although ∗socer would not be unexpected here).
- These mostly gave -trum in Latin. E. g... well, computatrum. :)
- uter, neuter, and many comparatives (citior, altior, fortior... I shall stop here).
Also, there is an interesting conference paper by Gorman³ discussing how certain words managed to possibly escape the rhotic change.
1. ver should have been etymologically *veror, and might have undergone rhotacism before shortening.
2. soror and uxor are not in this category, but they do indeed end in -r, so why not mention them? Also, mulier does not belong also, because the stem would originally end in s, later lost. Not clear why it escaped rhotacism, as the loss should pre-date it. Let's just do not count it in.
3. The link is directly to paper, so in case it disappears, here's the cication: Gorman, Kyle. "Exceptions to rhotacism." Proceedings from the Annual Meeting of the Chicago Linguistic Society. Vol. 48. No. 1. Chicago Linguistic Society, 2012.