[wo nuki ni shite/wa nuki ni shite] can have two generally the same usages. It can be used to show that something is without, or left out. Wa does not have to follow after shite but it does give extra emphasis as [wa] is frequently called the highlighting particle. [nuki] itself can be used to follow a noun directly to mean "without" especially when listing ingredients or talking about doing something "without". [Ni shite] itself gives meaning to nuki showing that it is "for" whatever [nuki] is excluding that causes/does such and such. So [nuki ni shite wa]could be looked at as meaning for without... I would also like to mention that a lot of people need to watch their Romanization on this site as somethings are just wrong. This page should be wo nuki ni shite/ wa naku ni shite not ha, wa is just a spelling rule since the Heian period to be written with the Kana character for ha.
#8080/1 are better translated as: Everyone, let's enjoy drinking without talking about (or, discussing) work. More natural sentence: Can we not enjoy drinking without talking about (or, discussing) work?
EvilKyra presented the fact that "nukinishite" means "leave off or leave out" how does that affect the English translations?
I think you don't need the ending "は", the actual grammar part seems to be 'nukinishite', which means 'leave off or leave out.'