Does a monster burrowing through stone or earth leave a tunnel?
For example could an Adult Blue Dragon just burrow 30 feet to be near invulnerable until it comes out? Or could a fighter jump down the hole it made?
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Consider the Earth Elemental:
Earth Glide: The elemental can Burrow through nonmagical, unworked earth and stone. While doing so, the elemental doesn't disturb the material it moves through.
The Earth Elemental (and the Xorn monster) can just glide through earth with its Burrow speed, without disturbing it. Monsters without such exception are assumed to disturb their environment.
Reading the Burrow rules:
A monster that has a burrowing speed can use that speed to move through sand, earth, mud, or ice. A monster can't burrow through solid rock unless it has a special trait that allows it to do so.
Monsters like the Purple Worm and Umber Hulk have the ability to burrow through solid rock, and specifically leave a tunnel behind them:
Purple Worm: The worm can burrow through solid rock at half its burrow speed and leaves a 10·foot·diameter tunnel in its wake.
Tunneler. The umber hulk can burrow through solid rock at half its burrowing speed and leaves a 5 foot-wide, 8-foot-high tunnel in its wake.
Others, like the Ankheg and your exemplary Blue Dragon, also leave tunnels behind them. However, it depends on the environment whether the tunnels remain, or immediately collapse.
Earthen Tunnels. As it burrows through earth, the ankheg leaves a narrow, partially collapsed tunnel in its wake.
Blue dragons make their lairs in barren places, using their [...] burrowing ability to carve out crystallized caverns and tunnels beneath the sands. A blue dragon will collapse the caverns that make up its lair if that lair is invaded. The dragon then burrows out, leaving its attackers to be crushed and suffocated.
The dragon clearly has the capacity to create standing tunnels, but also to burrow out while leaving no chance of escape to unlucky adventurers.
Whether monsters leave a clear tunnel behind them, or the ground collapses behind them to block their path is environment- and DM-dependent. I would rule that, in lose sand, the hole would be covered. In hard ground, a tunnel would be left behind. In lose mud (like a mountain side), rocks would collapse and cover the tunnel again.
Ask yourself one question: As the creature digs where does the dug up material go? The answer is usually right behind them. They are burrowing a hole to move themselves, not excavating. Meaning there are good odds the hole has filled in directly behind them.
Unless otherwise noted, one should assume the recently burrowed material ends up immediately behind the creature. There are monsters that have explicit caveats regarding their ability to leave a tunnel (like the Purple Wyrm and the Umber Hulk).
Sometimes Physics also plays a part in this too. An Adult White Dragon burrowing a hole straight upward into the roof of their ice cavern lair will have the dug out ice fall down into the Cavern. Conversely if the Dragon burrows laterally then the dug up left overs of the burrowing remain in the space directly behind the hole they burrow.
Each situation has the ability to be different and should be considered by the DM.