My constructed language forbids vowels from running consecutively. That is, there will always be a consonant between two vowels. Devised the rule on a whim.

Is it plausible? Can you think of any general principle of language which would prevent this from happening?

Yes, this is reasonably common. Luganda, for instance, does not allow two vowels in sequence; where such sequences would occur, it lengthens the second vowel and either deletes the first or converts it into a semivowel if the first vowel is high.

For a more direct correspondence to your strategy, some dialects of English insert /r/ between consecutive vowels.

Forbidding hiatus between consecutive words is also quite valid.

Ancient Greek was against vowel hiatus rather strongly, especially the Attic dialect. I mention it only because it's rather amusing how many different ways it invented to eliminate hiatus: elision of the first vowel, elision of the second vowel, contraction into a long vowel or diphthong, epenthesis of /n/, epenthesis of /t/ (arguably, from -mn̥-).

The Attic methods of forbidding hiatus between consecutive words are interesting.

I've also invented a sort of double 'a' vowel with the 'a' from 'apple' or 'attic.' So, 'jaarsis' would be pronounced 'jah-arsis.' Would the disuse of consecutive vowels make the 'aa' unlikely? Would such a construction naturally suffer elision into a simpler form?

That depends on what you mean by "double 'a' vowel". The way you transcribe it, it looks like you have a syllable boundary there, which means that unless you have an intervening glottal stop you'd be violating your own language's hiatus rule. If you mean something else, I highly recommend you take a look at this convenient transcription method to make it more clear.

By IPA, it's this sound: æ, or, as I'm using it: ææ pronounced one after the other. No glottal stop, in violation of my own hiatus rule. Is this sort of exception plausible? Am not sure why a language, like Luganda or Attic, would develop to disallow two vowels in sequence, so it's hard to envision why there might be an exception.

I haven't done any quantitative studies on it, but I imagine such a situation would indeed be quite rare. It depends, somewhat, on the phonological context of your language. Is hiatus avoidance something that there are synchronic phonological rules for? As in, do underlying sequences of vowels occur, which are then resolved somehow, or do they just happen to not occur for whatever reason? It could be, for instance, that your language lost some consonant specifically between two <a>, such as /h/ or /ŋ/.