(The answer from Midas is correct and more authoritative, here I will answer with a different focus.)
The current thinking on the classification of Hurro-Urartian is still that it is a primary language family, with no phylogenetic relation to Indo-European.
There were generally rejected attempts by Diakonoff to connect it to the Northeast Caucasian family of Lezgic and Vakh, which as far as I know were based on coincidences or loanwords or other features that one might expect after thousands of years of contact.
So there is no direct relationship. There are three main points of contact between Hurro-Urartian and Indo-European:
Contact between Hurro-Urartian and ill-fated Indo-European languages like Luwian and Hittite, which was a very dominant language across the region.
Contact between Hurro-Urartian and Iranic languages, over which little is written, but from the times and dates we know must have been significant. It includes Cimmerian from North of the Black Sea.
Contact between Hurro-Urartian and Armenian and proto-Armenian.
The latter was intensive. Armenia was originally a translation of Urartu, and Ararat - the Semitic and then European cognate of Urartu - is still generally synonymous with the Armenian highlands. Hurro-Urartian languages were still widely spoken in Armenia in the first millennium BC, the language shifted to Armenian only gradually. Unsurprisingly, the Hurro-Urartian substrate in Armenian is significant and mostly in those words for native concepts. There are also speculations about structural influence, certainly Armenian while typically Indo-European in some ways has areal features, for example lack of grammatical gender, and arguably post-positions, and phonology, for example consonant clusters.
Beyond that, there is some less direct contact between Hurro-Urartian and the ancestors of other living Indo-European languages, for example via Hittite with Greek, via Hattic or via Semitic, which was very dominant. Note that there is a growing view that Armenian, Greek and Albanian split a bit later than the rest, that Armenian was related to Phrygian, and that the proto-Armenian language moved eastward, but this is not the only view.
There are notably Wanderwörter from the region of unknown origin or considered simply substrate, for example cherry, and ancient loans like wine. But of course historically there has been too much focus on vocabulary and even pronunciation and strict trees and not enough on deeper mechanics and possible fusion.
There is also the Anatolian hypothesis. Personally I find it implausible but accept that the overall story may be complicated, clearly multiple waves of Indo-European languages were attested in the Armenian highlands and Mesopotamia even back when nearly all civilisational output was in the other direction.
The longer and more intense the contact, the more the languages bleed together, and the more tempting it becomes to purposefully or foolishly re-construct a pseudo-scientific phylogenetic relationship.