Is it correct that the word “Dune” comes from a very old Greek root?

A dune is a heap of sand. We can track it to Gaulish *dunom. Maybe.

A θίς can be a heap of several things, including sand.

A relation between the two has been suggested, but it’s not certain. To quote Frisk:

No satisfactory explanation. Wackernagel compares Old Indic dhíṣṇya– ‘situated on a knoll’, ‘knoll strewn with sand’, which could go back to an IE *dhisen-, *dhisn-. Often compared to German Düne ‘dune’ and related words, either as *θινϝ- related to Old Indic dhánvan– ‘dry land, mainland, beach’ (Fick; but that does not account for the /i/), or as *θϝιν- related to Lithuanian dujà ‘particle of dust’, cf. θύω ‘to storm in’. According to Osthoff, from Old Indic –dh-i in ni-dh-í ‘putting down, keeping’ (see τίθημι).

So… definite maybe.