I wasn't comfortable with some assumptions people are readily making in their comments, so I did some research and a few calculations. I also incorporated some suggestions made in the comments.
The crushing weight of the earth
He digs a hole big enough for himself (between 3-4 feet deep)
Lots of people claim he'd have trouble breathing, which is likely to be true, but let's get some numbers. Earth has a density as high as 2tons/cubic meter, so the following represents worst-case scenario.
If his hole is about 4ft deep, and assuming a character 6ft tall and 3ft wide occupying 1ft from the bottom (that is, there are 3ft of earth above him) he is under roughly 54 cubic feet of dirt, which is about 1.5cu.m, or 3 tons (metric). At 3ft, that's 2 tons—in fact, for the proportions I chose, for each foot of dirt above his body, there'd be a total increase of 1 ton in weight. But that's distributed over a rectangle covering his whole body, how would each body part suffer?
About 30% of that weight would be directly on his chest (assuming his torso is about 30% of his height). This means it'd be supporting and breathing against something between half a ton and one ton—or one or two horses. His neck would be under about at least 40kg, which is like a rottweiler standing on it—he might get a sore throat. His legs would be supporting about a horse each—unless he wants a gangrene that's a no-no.
Armour and other forms of protection
I assume your player wouldn't be sleeping naked. If he's wearing some form of rigid armour (so the weight is on the chestplate, not on his chest), could he bear the weight? Well, let's see.
According to this short article, a baseball bat swing can deliver a peak force of over 36 thousand Newtons. That's equivalent to 3.6 tons, so if you think the character's armour should be able to handle that without breaking, it should be a breeze. Considering that plate armour only dents against a war hammer swing, even though all that weight is being applied on a much smaller area, I think it's enough. He wouldn't, however, be able to move—that'd be a really uncomfortable night though, sleeping in full plate mail and unable to move. And I don't want to think what happens if he needs to go to the toilet.
There's a better way around it: if he was able to create a structure over his body able to support that kind of weight, that should protect him. I don't mean something as sophisticated as the shelter Brian suggested (though that's even better an idea), he can use stones or wood to build something more rudimental, like a wooden platform. That would force him to gather supplies to build it each night or dig shallower if that's unavailable.
If he can't build a structure like that, even with very resilient plate armour he'd be unable to move. Without a helm, he'd have his face caked with damp earth for the whole night—that's not reasonable, if the player disagrees ask if he can fall asleep with a wet towel on his face. You can ask him to devise a way to create such a structure and the means to transport it (going around dragging a coffin or something similar would completely make up for it, in my opinion), or bury himself less deep (0.5-1ft layer of dirt over his body would still be efficient at hiding him, easier to support and generally more believable).
Temperature and humidity
Some people mentioned the cold and dampness. That's a valid point. I'll start with the cold.
If it's decided he must have something that would protect him from the weight of the earth, I don't think cold would be a problem. Actually, being in a cramped space would also warm up the air around him. Snow caves are an example, suggesting that if he had some empty space, being underground would probably make him warmer than lying above the ground, where there might be wind or dew.
I couldn't find anything convincing about moisture below 3ft deep, but down to that point, it's fairly regular. I understand the logic, dampness is a problem in underground buildings—but generally, these lay a lot deeper than 4ft underground. I don't find anything to support the idea that it'd be a lot damper under 3 or 4 feet than it would on the surface—unless, of course, there has been a lot of rain, the party is near a body of water or above a shallow water table. If you have a map of the region, you could judge that from the position of water bodies. You could roll to see if he hits water—or rocks, for that matter.
Bottom line: Penalties to sleeping on damp surfaces may apply if he can't protect himself from dampness, I don't see the cold being a big factor. Frequent or intense rain could frustrate his plan.
Animals, insects and other creepy-crawlies
A number of animals live underground, a few of which could be dangerous to find. Several are carnivores that might attack to defend themselves if one runs into their lairs–badgers, for instance, are known for being fierce. I'm not sure if any would burrow so deep, but if they find something underground that looks defenseless, tentative bites might be possible.
Several others might be inconvenient in different degrees: ant colonies, especially fire or bullet ants, would suck to run into, and colonies can go quite deep. Tarantulas and spiders in general, centipedes and other carnivorous and/or poisonous insects might be even worse, but they tend to burrow in shallow trenches, so I'm not sure how much of a problem that would be during the night.
There are myriads of microorganisms that live in earth, especially when it's warm and damp. Several bacteria and fungi can cause dangerous diseases, and sleeping underground frequently would expose the character to them a lot more frequently.
Breathing through a straw
Breathing through a straw requires keeping your mouth close and doing
so while you're sleeping is impossible.
If he actually manages to create a chamber around his head, that'd be fine. He doesn't have to have the straw in his mouth to breathe through it. Actually, that'd be a terrible idea, as any weight falling on the straw from above could cause it to stab the character in the mouth, which may also give him a sore throat.
Another soil weight related complication: the straw itself should be able to support the same kind of weight as the player, or it'd collapse at the bottom. It's not impossible, just inconvenient: what is it made of? How was it made? How did the character come by it? How does he carry it around?
There are other important things to take into account: straw must have enough airflow to allow him to breath normally (which constrains the diameter) and the volume needs to be smaller than his lungs' (which constrains the total volume), if the end is exposed, particles or liquids might enter, and bugs might crawl in looking for somewhere warm.
Contrary to what has been said in other answers, the dimension constraints aren't too hard to meet. If we have a straw 0.5in internal diameter (which seems plenty, compared to tracheal tubes and nasopharyngeal airways) and 4ft long, the volume is about 6e-3 cubic feet, or about 0.17l. This is not much of a problem: human adult respiratory tidal volume (the amount of air pulled in or pushed out in calm respiration, which is more relevant than the lung capacity per se) is normally around 0.5l. That means that for each breath, 66% of the air pulled in is fresh, which is enough: oxygen concentration in atmosphere is about 21%, but exhaled air has about 15% oxygen. Therefore, 66% fresh air means he'd be breathing around
(21 * 66%) + (15 * 34%) =19% oxygen, which is enough to survive. For the level of oxygen drop below 16% (the threshold where trouble starts), his expiration would need to have less than 7% oxygen‚ which seems unlikely.
It could also happen that the straw gets pulled out from outside, causing his airway to collapse under it's weight. Would he suffocate? Adult humans breath at least 12 times per minute, and each exhalation has at least 4% carbon dioxide, which means about 20ml (assuming a 500ml Tidal volume as before). That adds at least 240ml, or 0.008cu.ft of carbon dioxide each minute. Oxygen is inspired at 21% and expired at 16% of less—therefore, depleted at 25ml/breath, translating to 300ml or 0.01cu.ft per minute. Low oxygen or high carbon dioxide cause accelerated breathing, but I'll ignore that for simplicity.
We had assumed an 18cu.ft hole. Let's say his equipment and his body occupy 1/3 of that. At the moment of the collapse, there are 12cu.ft of air, nearly no CO2 and 2.52cu.ft of oxygen. Carbon dioxide is not toxic, but starts causing mental confusion at a concentration of 6%. 6% of 12cu.ft is 0.72cu.ft, it'd take 90 minutes to reach that level at the rates calculated. After another hour, it reaches 10%, and after another hour in 10% or under, the character would asphyxiate. Oxygen, on the other hand, would be dangerous below 10% (or 1.2l). It'd be at that level after 132 minutes, and after 180 minutes it'd be below 6%(or 0.72l), causing the heart to stop. Whoops.
As suggested in the comments: if the DM woke up feeling funny, it might just happen that some animal spots the straw and, upon smelling strangers, decides to mark their territory. Then again, the character might just like it.
Interactions with the party
He has no intention of quickly being able to help the rest of the
party if they get attacked so escaping his hole is not something I can
use against him (I tried).
That's the biggest problem I see in his method. If you think any of the characters wouldn't be okay with it based on their traits, make that character justify why they're not doing anything about it. Otherwise, that's not your problem, but the party's. Perhaps everyone else is equally selfish in allowing this to get extra XP and loot, which might end up creating a big gap between him and other players, but I assume he realizes this.
The party would also need to be willing to dig him out every morning, and if he's using some form of platform, they'd have to help him build the thing every night, rebuild it if it's not good enough—that's a lot of work, why are they willing to help him do it? Will they have the energy or will to dig him out in the morning after a vicious fight? What if they were at risk of getting hurt in the process?
As someone else pointed out in the comments, there's another problem: digging a hole that big takes at least few hours. And that's assuming that they have a shovel or two lying around, which then brings the problem of carrying it. With no shovel, it's even worse, and getting hurt is unavoidable. Why is everybody willing to waste something like 10% of their day and a lot of energy to do this? ESPECIALLY before breakfast?
Personally, I wouldn't mind having a player like that in a party, as long as I didn't have to help him get in or out, nor carry shovels around. His survival entirely depends on each party member. If he ever crosses someone or if they got fed up, they can get back easily: by pouring something/farting down his straw, abandoning him buried, selling him out to the enemies. There are so many creative ways to get rid of or back at him that it's almost wasteful that I'd most likely just forget to dig him out.
How to deal with this as a DM?
Am I wrong to think that this is not such a great plan? I can't think
of any reason or mechanics to point out the flaws of his plan.
If you ask me, it's a not good plan because I wouldn't want to be in his position, but if he's not bothered and neither is the party, I don't get the problem. Again: if you think the party shouldn't be comfortable with this, force everyone to justify why they're going through pains just so the freak can have his way.
Otherwise, whether or not you, as a DM, should discourage this is a matter of opinion. Personally, I don't think it's the kind of thing the DM should have to intervene in—it's a problem for the character and the party alone. But ultimately, it's your table—you can kill him pretty easily if you want him to stop and he doesn't want to.
Regardless, one thing I think you could and even should do is roll every night for unpleasant effects. Some ideas:
- 10% chance of a perfectly restful night
- 30% for a small unpleasant effect: feeling uncomfortable all night, not getting enough sleep, having an itch, pissing himself, eating bugs, bugs crawling all over his body etc. He'd sleep, just not as much as everyone else.
- 30% for something more inconvenient: fire ants attacking his crotch for small damage, a mole stealing something shiny, water dripping on him all night, something keeping him from sleeping etc. No sleep tonight.
- 25% of something really bad: partial blockage of the straw so he can't breathe properly, an animal chewing or otherwise damaging the straw, liquids, dirt or mud seeping in the straw to slowly fill his "cave", an animal decides to see what he tastes like, some nasty disease etc. Perhaps some direct damage.
- 5% chance of something fatal or incapacitating: a disastrous collapse of his platform, somebody falling on top of the straw causing it to stab him, partial collapse causing a limb to be deprived of blood, potentially leading to gangrene/amputation etc.
I just made up these probabilities and scenarios, but I'm not familiar with dnd-5e. You can play with them based on the character, the party, their abilities and traits etc. Be creative with the unpleasant effects, think of cumulative ones (sleep deprivation, diseases) and so on and so forth.