As for 1-categorical exact functors, there is a general definition of exact functors that restricts to the simple conditions that finite (co)limits are preserved in the case that these exist.
For a regular cardinal, an (∞,1)-functor is -right exact, if, when modeled as a morphism of quasicategories, for any right Kan fibration with a -filtered (∞,1)-category, the pullback (in sSet) is also -filtered.
If then we just say is right exact.
This is HTT, def. 188.8.131.52.
If has -small colimits, then is -right exact precisely if it preserves these -small colimits.
So in particular if has all finite colimits, then is right exact precisely if it preserves these.
This is HTT, prop. 184.108.40.206.
This is HTT, prop. 220.127.116.11.
Section 5.3.2 of