Are there limitations on taking off or landing if airport lighting such as the VGSI (PAPI) is out of service?
There wouldn't be any limitation on taking off with the VASI/PAPI out of service, unless it involved a requirement to return to the airport where the VGSI would be required (see below).  That would be a very extreme case, because it would mean that 


the rules + the existing weather require the ability to return to the departure airport (i.e. a "takeoff alternate" is either not authorized or none is available), and
no other runway (including the same strip of concrete, landing in the opposite direction) is available at the same airport (tailwind component could cause this), and
this is a case (discussed below) where a VGSI inop prevents an approach to 'this' runway.


That's going to be a very rare combination.

So, when can an inop VGSI prevent an approach?  There are cases where US TERPS rules do require an operative VGSI in order to conduct certain operations -- mostly at night.  I've seen plates where a VGSI is required for circling operations to a given runway at night, and in one case where you couldn't fly any instrument approach, even an ILS, to the particular runway without the VGSI.  

The latter fell in the category of TERPS rules that don't make intuitive sense... if I'm flying the ILS glideslope, by the time I'm going visual I'm so close to the runway that you'd think obstacle clearances have all been met &amp; nothing remaining (in front of the aircraft) could possibly be a factor.  It was a memorable discussion because it seemed to make so little sense.  But, the restriction was published, and without the VGSI that runway wasn't usable.

The general case, however, is that the VGSI can be inop with no effect on the ability to fly an approach.  In 99.9% of the cases, it won't matter.  But for an obstacle in the right spot, you can get the note on the approach plate stating the requirement, and in that case, it matters.

But it's very uncommon.