I've been reading through Lurie's book on higher topos theory, where he develops the theory of $(\infty,1)$toposes, which leads me to the following question: Is there any sort of higher topos theory on the more general $\omega$categories, where we don't require all higher morphisms to be invertible?

The short answer is no. Even 2toposes are poorly understood  we don't know what the right definition is. For higher dimensions, including $\infty$, we definitely don't have the answers. Just as the primordial example of a (1)topos is $\mathbf{Set}$ (the 1category of sets and functions), the primordial example of a 2topos should be $\mathbf{Cat}$ (the 2category of categories, functors and natural transformations). Mark Weber has done some work on 2toposes, building on earlier ideas of Ross Street. But I think Mark is quite open about the tentative nature of this so far. There was a good discussion of the current state of 2toposes (and more generally ntoposes) at the $n$Category Café last year: 


One should keep in mind that Jacob Lurie's book "only" (if I may use this word) discusses the (oo,1)version of Grothendieck toposes/category of sheaves: the (oo,1)toposes in Jacob Lurie's book are (oo,1)categories of (oo,1)sheaves/of oostacks. This is less general than the "elementary (oo,1)toposes" that one would eventually want to see, but it already goes a long way  and it is more accessible. Similarly, while a general theory of ntoposes for higher n is largely missing, there is a bit more known about (oo,n)sheaves, i.e. of oostacks which are presheaves with values not just in oogroupoids but in (oo,n)categories. For instance Ross Street once proposed a notion of descent for strictomegacategoryvalued presheaves. Using a result by Verity this may be regarded as presenting descent for strict oogroupoid valued presheaves, but I'd expect that with the required care exercised it goes further than that (and this seems to be what Street had in mind, though I can't tell that for sure). But a more developed general theory for descent of (oo,n)category valued presheaves is developed notably in HirschowitzSimpson's Descente pour les nchamps. This yields at least part of a theory of Grothendieckstyle ntoposes. 

