This seems to be quite the knotty little tangle of roots, so this is going to be rather long.
LIV² gives a total (that I’ve found) of seven different roots for this rather complex mass of running and sleeping:
- ?*derdʰ-, ?*dreh₁- and 2. *drem- ‘sleep’
- *dreh₂-, 1. *drem-, *dreu̯- and ?*du̯er- ‘run’
(Question marks indicate that the existence of the root is uncertain, usually because it’s based on very few attestations within just one language.)
Sleeping: ?*derdʰ-, ?*dreh₁- and 2. *drem-
The only forms quoted directly under ?*dreh₁- ‘sleep’ are Vedic: an optative ni-drāyā́t, a participle -drāṇá- and a future drāsyáti. This root is apparently quite rare in Vedic, perhaps, they surmise, because of its homophony with the ‘run’ root(s). The only comment on the shape of the root is:
Ansatz mit *h1 aufgrund des viell. mit 2. *drem- (s. d.) kontaminierten slav. drěm-, das aber auch anders erklärt werden könnte.
Under 2. *drem-, they give the Latin forms (which yield the Romance forms) as well as Old Church Slavonic ‑drěmljǫ ‘slumber’, which underlies the Russian form cited in the question. They say very little about the actual shape of the root, except to note that the OCS forms are perhaps the result of a contamination between *dreh₁- and 2. *drem- > *dreh₁m-. This in turn is why they posit *h₁ instead of *h₂ or *h₃ as the final consonant above: if it had been one of the others, the contaminated form would have resulted in a different vowel (*-dramljǫ I think, but Slavic is not my forte). They hedge their bets a bit, though, saying that the Slavic form may also be
mit Klingenschmitt, FS Meid 819, als innerslavische Intensivneubildung zu erklären.
So in conclusion: there is a PIE root 2. *drem- meaning ‘sleep’, and also a rarer root *dreH- meaning ‘sleep’. If the Slavic verb is indeed a contamination of these two roots, *dreH- should specifically be *dreh₁-; if it’s not, we don’t know which laryngeal it is.
The third root, hypothetical ?*derdʰ-, is based only on the Greek verb δαρθάνω ‘sleep’ and its forms (e.g., aor. ἔδραθε). It is possibly an extension invented within Greek.
All three roots are clearly connected, but their internal connection is hard to pin down. This is not uncommon – there are quite a few cases in PIE where looks like a common ‘base root’ is variously extended with an extra consonant for seemingly no systematic or meaningful reason. There is no evidence of the base root itself (**der, presumably) actually existing.
Running: *dreh₂-, 1. *drem-, *dreu̯- and *du̯er-
In the case of running, there are at least four connected roots – presumably extensions to an underlying, but unattested, base root of the same shape as the ‘sleep’ root. Unfortunately, at least one of the extensions is also identical to one of the ‘sleep’ extensions (*-m-), which, frankly, is just PIE speakers making things difficult for themselves (and us).
The m-extension 1. *drem- only appears to be attested in Greek and Indo-Iranian (at least those are the only forms quoted in LIV²). In Greek, the suppletive aorist and perfective stems of τρέχω (and sometimes θέω) ‘run’ is aor. ἔδραμον, perf. δέδρομα – this is where δρόμος comes from. From IIr. are quoted Khotanese dremäte ‘drives off’ < causative *drom-ei̯e-, as well as an intensive form from the Kaṭha Upaniṣad, dandramyamāṇa- ‘running around, milling about’. Based on these, they give the translation ‘run (somewhere)’ (“(wohin) laufen”) for this root, though I’m not sure exactly how this really differs from just ‘run’.
This root would be completely homophonous to 2. *drem-, but they would not necessarily be likely to be mixed up. As far as attested forms show, the ‘sleep’ root formed only a present stem (*-i̯e/o-), whereas the ‘run’ root formed only a (root) aorist, a (reduplicated) perfect and a causative. While they had the same shape, their formations were thus in complementary distribution.
The laryngeal extension is in this case certainly reconstructible as being *h₂, based especially on the Greek present, -διδράσκω < *di-drh₂-sḱ-. The meaning of *derh₂- seems to have been a little more specific, referring to running away: the Greek form is only attested in compounds, nearly always ἀποδιδράσκω ‘run away’, and some of the Indic forms clearly mean ‘run away’ as well, even with no prefix.
If the Slavic ‘sleep’ forms are actually contaminations (see above), that would mean that the ‘run’ and the ‘sleep’ roots would be phonetically different in PIE, though they would of course merge once the laryngeals were lost. Even if they were entirely homophonous, though, they would still not coincide, because – as was the case with *drem- – their formations differ: 1. *dreh₁- ‘sleep’ forms only a root present, while 2. *dreh₂- ‘run (away)’ forms only a (root) aorist, a *-sḱe/o- present, a (reduplicated) perfect and a desiderative. The only stem they both form is the present, and they form that in different ways (root present *dreh₁-ti vs *-sḱe/o- present *dṛh₂-ské-ti).
The last two roots are a *-u̯- extension *dreu̯- and what looks like a metathesised version of this, *du̯er-. Both are only attested in IIr. (the former seemingly only in Vedic, the latter only in Avestan, at least as verbs). At any rate, these (like the Greek ?*derdʰ- ‘sleep’) need not concern us here, because they’re clearly distinct from each other and cannot be confused.
As far as the evidence I have available can tell us (primarily the OED/Etymonline, Brill’s etymological dictionaries of Latin, Greek and Germanic, LIV², plus of course Wiktionary), it seems PIE had two different base roots of the exact same shape, used as the basis for various extended roots (all of which seem to mean more or less exactly the same thing as each other), of which at least two (possibly four) were also completely identical to each other in form.
Within PIE itself, however, there was no chance of confusion, because the one (or two) set(s) of roots that had identical shapes for different meanings formed different stems in different ways. In practice, it would thus always be clear which verb you were using.
So I would say the Wiktionary wording “*drem- (‘run, sleep’)” is an oversimplification. It wasn’t a matter of having one verb that meant opposing things – though this is synchronically entirely possible (cf. English cleave ‘cut apart / bind together’ or sanction ‘allow / penalise’) – but of having two words that happened to have the same form in their root structure.
Whether these two roots originated as dissimilations from the same underlying base root (i.e., whether there was originally one or two *der- roots to extend from) is impossible to say. Semantic shifts of a comparable scale to ‘sleep’ > ‘run’ (or vice versa) are not unheard of, so it could have happened – just think of how wicked now also means ‘excellent’ in English – but there’s no particular reason to believe that these aren’t just two roots that happened to have the same structure.