Is there a minimum amount or type of lighting, in the form of runway or approach lights, that must be present for a runway to have an instrument approach? I'm curious to know in the context of both the FAA and ICAO.
I have no idea about ICAO but as far as I can tell from the FAA's TERPS, there's no general requirement to have any runway or approach lighting at all (see below for an example), although some specific approach types (like CAT II/III ILS) do require it. Where there is lighting, it's used to reduce the visibility minimums, but the calculations assume no approach lighting by default.
This is from 3-1-2(c) (emphasis mine):
Approach lighting systems extend visual cues to the approaching pilot and make the runway environment apparent with less visibility than when such lighting is not available. For this reason, lower straight-in (not applicable to circling) visibility minimums may be established when standard or equivalent approach lighting systems are present.
The TERPS concept seems to be that the approach minimums can be reduced depending on the lighting type available. Section 3-3-2 has an example of calculating visibility minimums when there are no approach lights available.
As for a specific example, MDO/PAMD (Middleton Island, AK) has three approaches (2 RNAV (GPS), 1 VOR) but - as far as I can tell - no lighting of any kind. The chart supplement says:
Rwy 02–20 marked with rwy cones. Rwy 13–31 marked with cones and OTS lgts. Many missing.
As you might expect, the approaches are not authorized at night :-)
Finally, part 121/135 passenger flights can't land at night unless the runway is lit (121.590, 135.229), so any airport that handles those flights will have runway lighting at least, and almost certainly approach lighting too.
This is a pretty complicated question (mostly because it involves the 312 page FAA Airport Lighting Design AC 150/5340-30H).
For example in regard to edge lights, from page 3:
2.1.1. Selection Criteria
The selection of a particular edge lighting system is generally based on the operational needs per the following guidelines:
LIRL - install on visual runways (for runways at small airports),
MIRL - install on visual runways or non-precision instrument runways,
HIRL - install on precision instrument runways,
MITL - install on taxiways and aprons at airports where runway lighting systems are installed.
As stated, the above are general selection criteria. However, the airport surface requirements for specific approach procedures are the determining factor for system selection. See AC 150/5300-13, Airport Design, for more guidance.
It goes on to describe the different types of lighting required for MIRL/HIRL. It really goes into detail about when centerline lights, guard lights, stop bars, clearance bars, edge lights, threshold lights, etc are required for different types of runways. It goes into quite a bit of detail about what type of light is required and when.
It is important to note that these are Advisory Circulars, not FAR's. Airports really don't need to be anything more than a mowed lawn. The FAA has an airport certification process that uses these AC's as guidance for "good airport design", but the design may be modified to fit certain situations. These are only to be used as guidance in the design/construction of an airport.