From Middle English day, from Old English dæġ (“day”), from Proto-Germanic *dagaz (“day”), from Proto-Indo-European *dʰegʷʰ- (“to burn”). Cognate with West Frisian dei (“day”), Dutch dag (“day”), German Tag (“day”), Swedish and Danish dag (“day”), Icelandic dagur (“day”). Compare Albanian djeg (“to burn”), Lithuanian degti (“to burn”), Tocharian A tsäk-, Russian жечь (žeč’), Sanskrit दाह (dāha, “heat”), दहति (dahati, “to burn”). Latin diēs (from Proto-Indo-European *dyew- (“to shine”)) is a false cognate.
The standard reconstruction identified three coronal/dental stops: */t/, */d/, */dʰ/. They are symbolically grouped with the cover symbol T. In so-called "thorn clusters" of the form TK in all branches except Anatolian and Tocharian a metathesis occurred, resulting in dorsal-coronal clusters of non-obvious phonetic makeup. Metathetized and unmetathetized forms survive in different ablaut grades of the root *dʰégʷʰ "burn" (whence also English day) in Sanskrit, dáhati "is being burnt" < *dʰégʷʰ-e- and kṣā́yat "burns" < *dʰgʷʰ-éh₁-. See the section on PIE phonological rules, below, for more discussion and examples.
I doubt this etymology just because I can't find any reference of this kind of saying that PIE *dʰegʰ- "day" and *dʰegʷʰ- "burn" are cognates.
PS: the reconstructions are based on here, in footnote 8 on page 2 of which the author even deny this connection and E. day comes from PIE *dʰegʰ- "to repeat itself (over and over again); cycle".