Based on what’s written in the Oxford Dictionary, it’s context-dependent: ‘any’ refers to either a single or some amount of whatever is referred to; in other words I would argue one could benefit by treating ‘any’ as an adjective (even though it is a pronoun, determiner or adverb). I’ll allow myself to use the examples given by OD (punctuation added):
- We needed more sugar but there wasn’t any left. (singular verb)
- Are any of the new videos available? (plural verb)
In the first case, ‘any’ refers to ‘sugar’, which is singular; hence a singular verb is required. Note that ‘sugar’ is singular, despite the phrasing being ‘more sugar’; this can be read as ‘more of sugar’, which reveals that we are dealing with a partitive genitive; with partitive genitives, that which is being referred to is usually singular. In the second case, ‘any’ refers to ‘the new videos’; it is clear then that the verb needs to be in plural.
Referring back to your example sentences—‘Does any group want to give their presentation?’—one could analyse it the same way: From context, it is clear that any is referring to a singular; we could have written it as ‘Does any one group want to …’, making it clear that group should be treated as singular. Were the question phrased differently, e.g. ‘… any [of the] groups …’, then ‘any’ would (again, like an adjective) be pointing to the plural ‘groups’, requiring the plural verb: ‘Do any [of the] groups …’.
All the above disregards that one could consider ‘group’ a mass noun, thereby—as has been pointed out by the others—allowing it to take a plural verb. The method for analysing and understanding the syntax still applies, though.
Source for grammar: Eitrem, Latinsk grammatikk, 3rd edition (Tosterud & Kraggerud), Aschehoug 2006: §§ 81 and 82.