Summary: A door is a tactical (sometimes, strategic) element with plenty of uses and a cost to manipulating it. Break it if you are in a hurry and enemies know of you, or if there are no enemies, while leave it functional if there is a chance of using it against your enemies.
Chests are similar but less complicated.
Breaking doors makes noise
Noise alerts creatures. The usual rate of wandering monster checks in large dungeons with mobile inhabitants is, for me, once every ten minutes, with 1/6 chance of encounter. I also roll after noise occurances, such as combat or breaking down doors. And, of course, any nearby creatures will often hear the noise automatically or with good chance. If breaking in takes significant time, that also triggers a wandering monster roll or moves the next one closer.
This is not an issue in places without wandering inhabitants, such as very small dungeons or ancient sealed tombs.
A broken door is a sign of forced entry
This is an issue in civilized or organized dungeons, such as cities and fortresses, or when trying to rob the ancient mage's tower without the ancient mage figuring it out.
Dead enemies everywhere is also a sign of forced entry, so this might not matter too much in many situations.
Doors control the movement of others, too
Wolves and unarmed zombies will have very hard time getting through sturdy wooden doors, even given time. Many enemies will spend at least a moment to penetrate a door, and might not be clever enough to open one. Doors can be barred for further security. In a typical dungeon environment with some animals or weaker creatures, doors can be used for good benefit.
Also, if one wants to rest within a dungeon, closing and barring a door creates a nice alarm system and might prevent some wandering monsters from reaching you at all.
Doors control vision (and sound a bit)
Controlling vision allows hiding and resting and makes ranged attacks a lot harder. Hiding behind a door to run away or surprise someone is a classic maneuver.
Not everything breaks doors
A rondel or a sword is not the proper tool for breaking a door - you would get a dull edge or a small hole, which is often not very useful. It would be a waste of a battle axe, too - use a woodcutter's axe or something similar. The rules of D&D 5 leave simulating such concerns on the dungeon master, whose duty it is to keep the game world credible. (We prefer to share this duty among the entire gaming group.)
The matter of chests
Chests protect things. Breaking a chest might break those things, or you might want to have a nice chest for other purposes, if it can be carried. The concerns of noise are there, as are leaving tracks.