I don't think I've ever seen the books mention much, if anything, about how ceiling height affects a party in a dungeon. I'm preparing for my party to enter a dungeon with considerably low ceilings, and I want to make sure I'm not missing something.

Essentially, I have some very tall PC's going into a dungeon that has a ceiling height lower than their max height. Should this affect movement speed, combat, or anything else? I'd imagine at a certain point, the party will encounter some drawbacks to being in a 5-ft wide dungeon with a very low ceiling, but I don't know what the drawbacks would be.

See PHB, Chapter 9: Squeezing into a Smaller Space.

Move at half speed, disadvantage on attack rolls and dex saves. Attacks against them have advantage.

Edit: as scatterbraind said, if you are size Medium, a 5 foot ceiling is fine with no penalties.

Just note those rules are for fitting into a smaller space than your size category, which for all PCs is a 5' cube. RAW a Goliath suffers no penalties in a room with a 5' ceiling.

You are free to add some, but the squeezing rules are pretty harsh and I would be upset as a player if I was at such a disadvantage for an entire dungeon just because I picked the wrong race.

Unless you decide to impose a movement penalty, the main problems would be that they have no room for re-positioning and have difficultly avoiding traps even if they know they are there.  

Any races with racial Flight are going to be very upset.

The absence of a PHB/DMG  rule does not mean there is no penalty, it just means it is up to the DM.

Personally I think you are freaking fool if you do not give penalties for moving through a space that you clearly can not fit in without bending down.

I would rule you have to make an acrobatics check DC = 10 each round you are in a space smaller than your height.  Failure means you move at 1/2 speed and have disadvantage on any physical action.

depends how low compared to pc height it is.  If you have a 6' fighter trying to swing a greatsword around disadvantage makes sense.  However if they were using a thrusting type of weapon id allow it to roll normally.  I wouldn't impose the squeezing rules on the players though since its too harsh and as someone stated there are already some pretty nasty consequences of navigating a tight space.

I would say impose extremely harsh penalties on the taller races. Mostly to educate your players about how awesome halflings and gnomes are. Impose the harshest penalties on elves, because they are awful people. Watching an elf being forced to stoop is like watching an arrogant cat fall off a fence... Delightful!

Joking aside, I don't see the problem with light penalties on the larger characters. Smaller PCs already have disadvantages with jump height and vertical reach, after all.

Limiting the party to a single-file column through the dungeon (and reading up on the cover and concealment rules!) will already be a surprisingly big limitation on the party that is sure to give them a hard time. Being a stickler for rules about movement, line of site, etc. can already do a lot to set the theme without straying outside of RAW. But if you want to go beyond that...

5E doesn't generally hand out situational static +/- modifiers to attacks, favoring the advantage/disadvantage system. But that doesn't mean they don't still work well in limited situations, especially if it's clearly a one-off scenario that isn't going to have ripple out effects to the rest of your campaign. You might consider a -1 to hit for slashing and bludgeoning weapons, -2 to hit if they're 2H weapons, representing the reduced clearance for swinging. In a tunnel, shortswords, daggers, spears, and rapiers are king! 

You might also just give the medium party members (5' feet is pretty short for humans as well, not just goliaths) a -1 on Dexterity checks and saves?