Lamps have been around in the Old World since the Palaeolithic (one was found in the Lascaux cave). Candles are thought to have originated in the Bronze Age, but were certainly used by ancient Greeks, Romans, and Chinese within the 1st millennium BC. Were they used in the Americas before the arrival of Columbus?
Lamps were certainly used in the arctic, where they were known as Kudlik. The arctic peoples were of course late arrivals to America.
Candles of a sort were used by the northwest coast peoples. A local fish called the Eulachon, but known more colloquially as the candlefish, contains a lot of oil. The locals simply inserted a wick into a dried fish, and lit it. There are some videos of people lighting fish.
Apart from these two examples, I have not found any references to candles or lamps in early America. It is always harder to show an absence of course. Is anyone out there aware of examples? Otherwise I guess people just huddled around their fires at night…
As you say in your question, "It is always harder to show an absence" but in this case the absence of evidence, aside from the Arctic & Subarctic regions, seems complete. In short,
- none of the academic sources I've consulted mention the existence of pre-Columbian lamps or candles, but other means of artificial light are mentioned.
- examples of pre-Columbian art showing torches are easily found, but there are none showing lamps or candles.
- googling has turned up no hits for pre-Columbian Maya, Aztec, Inca, Native American etc. lamps or candles / candle holders whereas there are (often numerous) examples from early Rome, Judea, China, India etc., as well as post-Columbian lamp / candle artefacts in the Americas.
- there is one early 17th century source which explicitly states that, in at least one pre-Columbian Indian culture, candles and lamps were introduced by the Spanish.
- the very few sources which do mention pre-Columbian lamps have either misused a source or provide no credible evidence and / or belong firmly in the pseudohistory department.
Given the vastness of the area in question (two continents), the huge timescale and the numerous cultures involved, we cannot say with 100% certainty that lamps and / or candles never existed in pre-Columbian America but, if they did, the lack of evidence strongly suggests that their usage was definitely not widespread.
The only evidence for lamps in pre-Columbian America is in the arctic / subarctic regions. The article Litnik Archaeology at the Salmon Bend Site mentions several examples of oil lamps and the National Museum of the American Indian site has an example of a Kachemak lamp (see image below). More details can also be found in Encyclopedia of Prehistory: Volume 2: Arctic and Subarctic.
Stone Kachemak lamp, AD 500-1100, Cook Inlet, Alaska. Image source: National Museum of the American Indian
The only evidence which clearly supports the absence of candles and lamps in at least one region comes from the Spanish chronicler Antonio de Herrera y Tordesillas (1549 - 1626) who
tells of a wise Indian who, when asked to name the most important things he and his fellows had received from the Castilians, put chicken eggs at the top of his list, because they were plentiful, “fresh every day, and good cooked or not cooked for young and Old.” (The other items on his list were horses, candles, and lamps.)
Cited by A. W. Crosby in 'The Columbian Exchange' (30th anniversary edition, 2003)
Other 'evidence' mostly relates to its absence. For example, Jacques Soustelle in Daily Life of the Aztecs (1961), says:
resinous torches of pine-wood (ocotl) were used indoors, and outside links and huge braziers piled with resinous wood served for public lighting when circumstances -- a religious ceremony, for example -- called for it.
link = a torch of pitch and tow for lighting the way in dark streets
brazier = a container for hot coals, generally taking the form of an upright standing or hanging metal bowl or box
Charles C. Mann, in 1491: New Revelations of the Americas before Columbus mentions only torches for 'Native Americans'. Michael E. Moseley's The Incas and their Ancestors: the Archaeology of Peru (2001) also only mentions the use of torches, likewise Burr Cartwright Brundage in Empire of the Inca (1963). The Journal of Anthropological Research article A Reappraisal of Ancient Maya Cave Mining (2006) only mentions candles in a modern context, and it states that torches were used in mines. The website ancientoillamp mentions oil lamps from many cultures across the world, but none are from the Americas.
The reference cited in drewbenn's comment seems is at best inconclusive and probably wrong if this translation of the poem, 'Prayer to the Sun', the author cites is correct. The other source cited by drewbenn looks more promising but is far from conclusive (as Orange comments).
The idea that the Seneca's easy access to oil meant they must have used lamps can also probably be discounted:
the Seneca tribe, part of the Iroquois nation, collected seep oil for hundreds of years, using it as a salve, insect repellent, and tonic. Europeans called the dark, gooey substance Seneca Oil and found it effective for treating sprains and rheumatism. It also burned, but was unappealing as a lamp oil due to its unpleasant odor and smoke.
Also, there is no archaeological evidence to support the use of oil lamps among the Seneca. Gary Prost & Benjamin Prost's The Geology Companion: Essentials for Understanding the Earth also mentions the Seneca, saying they used oil for body paint and medicine (but no mention of lamps).
On the items on auction sites mentioned by Denis de Bernardy in his comment, googling has turned up nothing convincing. Related to this, in The Maya Indians of southern Yucatan and northern British Honduras (1918), author T. W. F. Gann mentions a 'small soapstone lamp' find but concludes that it is post-Columbian as the style is "totally unlike" that of ancient Maya culture. Images of post-Columbian lamps and candle holders are easily found (see here, for example)
(all emphasis is mine)
Similar objects found in Iceland have been interpreted as oil lamps
… is how this article describes an artifact from the L'Anse Aux Meadows site (in modern Newfoundland, Canada) and probably dating from around 1000 CE. Hardly a smoking gun, but it's a datum.