Example:
The dog climbed the fallen tree, then slipped and entangled his chain on/in/with it.
What's the right choice?
"The dog was entangled in his chain" The dog was caught in the chain
"The dog was entangled with his chain" The dog and his chain were entangled, possibly in something else, possibly in one another - however, the latter makes no sense since the chain cannot be entangled in the dog.
"The dog was entangled on his chain" The dog was entangled in something, and positioned over his chain
In your example: "The dog climbed the fallen tree, then slipped and entangled his chain on/in/with it." It seems that you want to say that the chain got caught in the tree rather than entangled: "The dog climbed the fallen tree, slipped, and his chain got caught in the tree".
To say that the dog was entangled with the chain:
The dog climbed the fallen tree, then slipped and became entangled with his chain.
To say that the chain was entangled with the tree:
The dog climbed the fallen tree, then slipped and entangled his chain with the tree.
Entangle is typically used with "in", "with", or "into". It can also be used as, "to become entangled".