Jelly, 3 bytes

ŒḊ’

A monadic Link accepting a list representing the tree: [root_value, left_tree, right_tree], where each of left_tree and right_tree are similar structures (empty if need be), which yields the height.

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How?

Pretty trivial in Jelly:

ŒḊ’ - Link: list, as described above
ŒḊ - depth
 ’ - decremented (since leaves are `[value, [], []]`)

Python 2,  35  33 bytes

Thanks to Arnauld for noicing an oversight and saving 4.

f=lambda a:a>[]and-~max(map(f,a))

A recursive function accepting a list representing the tree: [root_value, left_tree, right_tree], where each of left_tree and right_tree are similar structures (empty if need be), which returns the height.

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^{Note that [] will return False, but in Python False==0.}

JavaScript (ES6),  35  33 bytes

Input structure: [[left_node], [right_node], value]

f=([a,b])=>a?1+f(f(a)>f(b)?a:b):0

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Commented

f = // f is a recursive function taking
([a, b]) => // a node of the tree split into
 // a[] = left child, b[] = right child (the value is ignored)
 a ? // if a[] is defined:
 1 + // increment the final result for this branch
 f( // and add:
 f(a) > f(b) ? a : b // f(a) if f(a) > f(b) or f(b) otherwise
 ) //
 : // else:
 0 // stop recursion and return 0

Haskell, 33 bytes

h L=0 
h(N l r _)=1+max(h l)(h r)

Using the custom tree type data T = L | N T T Int, which is the Haskell equivalent of the C struct given in the challenge.

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Perl 6, 30 bytes

+$_&&1+max map &?BLOCK,.[^2]

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Input is a 3-element list (l, r, v).

C, 43 bytes

h(T*r)r=r?1+(int)fmax(h(r->l),h(r->r)):0;

Structure of binary tree is the following:

typedef struct tree

 struct tree * l;

 struct tree * r;

 int v;

 tree;

Scheme, 72 Bytes

(define(f h)(if(null? h)0(+ 1(max(f(car(cdr h)))(f(car(cdr(cdr h))))))))

More Readable Version:

(define (f h)
 (if (null? h)
 0
 (+ 1 
 (max
 (f (car (cdr h)))
 (f (car (cdr (cdr h))))
 )
 )
 )
)

Using lists of the form (data, left, right) to represent a tree. E.g.

 1
 / 
 2 3
 /
 4 5

is represented as: (1 (2 (4 () ()) (5 () ())) (3 () ())

(1
 (2
 (4 () ())
``` (5 () ())
 (3 () ())
)

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Charcoal, 29 bytes

⊞θ⁰⊞υθＦυ«≔⊕⊟ιθＦΦι∧κλ⊞υ⊞Ｏκθ»Ｉθ

Try it online! Link is to verbose version of code. Temporarily modifies the tree during processing. Explanation:

⊞θ⁰

Push zero to the root node.

⊞υθ

Push the root node to the list of all nodes.

Ｆυ«

Perform a breadth-first search of the tree.

≔⊕⊟ιθ

Get the depth of this node.

ＦΦι∧κλ

Loop over any child nodes.

⊞υ⊞Ｏκθ

Tell the child node its parent's depth and push it to the list of all nodes.

»Ｉθ

Once all the nodes have been traversed print the depth of the last node. Since the traversal was breadth-first, this will be the height of the tree.

Japt, 9 bytes

Ω¡ÒßXÃrw

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