Are there any light sources in "general" for the regions not otherwise lit by torches/lanters
Yes, but these are rare (phosphorescent moss, glowing crystals, etc.) and typically only provide dim light in a small (maybe 10 feet) radius--but sometimes they might be stronger.
Does anything emanate light or is it pitch black and your party needs infravision or a light source?
As mentioned,
mostly pitch black. So, PCs need a light source (torch, lantern, continual flame). Hooded lanterns are the most popular with the ability to reduce or expand the light available.
If someone has a torch up, do you force them to figure out if they are still going to hold the torch during a fight?
Oh yes! It means not having a shield so it is a big hinderance to the PC holding it. If they are a caster, they can't have a weapon in their other hand while holding a light source if they want to cast spells with somatic components (without War Caster feat anyway).
Or do you handwave darkness as not an interesting part of the game?
Never. Darkness is one of the thematic elements that is great for creating atmosphere and tension. Not being able to see what lies beyond your light source creates the unknown and personally I feel it pity for DMs who ignore this powerful tool.
We played yesterday and it makes for a significant issue when the PCs were moving through a goblin lair.
Now, I should clarify house-rules:
1. Nearly all PC races with darkvision no longer have it. I did a poll a while ago because something like half of more of the PC races have darkvision, which is ridiculous IMO.
According to my quick run of PC races I know of (I am sure I might have missed some...), more than half of all PC races get darkvision. IMO, this is a shame because darkness is one of the greatest elements for creating mystery and suspense, etc. For exploration, it is also a key component. And...
www.enworld.org
So, for PHB races, we now have Dragonborn (60'), Tiefling (60'), Half-Orc (30'). Elves have
shadowsight (dim light is bright, doesn't help in darkness) (60') and Half-Elves have 30'. All subraces with
superior darkvision keep it.
2. Dancing Lights and Lights are 1st level spells, not cantrips. No more infinite light sources (short of
continual flame, which at 50 gp represents hundreds of days (or nights) worth of light, but finding ruby dust to cast it is controlled by the DM (well, usually me...

). Again, using a hooded lantern for the continue flame is commonly used. We have had PCs put it on weapons (quarterstaffs with a leather cover you could place or remove as needed) or swords (so light would be available when the weapon is drawn), allowing them to carry their weapon and light source.